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Nascar threw a bash at Kansas Speedway in October to thank Sprint Corp. for being stock-car racing’s top sponsor for 13 years. More than 800 Sprint employees received hot dogs, burgers and seats to a nail-biting race.

One thing was missing: a new sponsor. Despite knowing for two years that Sprint was leaving, Nascar didn’t announce a replacement until December, when it announced that energy-drink maker Monster Beverage Corp. had won naming rights to the top-tier racing circuit.

Monster paid about $20 million, below Nascar’s asking price of $35 million and nowhere close to the original goal of $100 million, according to television and racing-industry executives familiar with the new contract. A Nascar spokesman wouldn’t comment.

With the first big race of the new season set for this Sunday, Nascar’s problems seem to have spun out of control.

About a decade ago, the sport was a cultural icon and inspired the hit car-racing comedy movie “Talladega Nights,” starring Will Ferrell. Since 2005, Nascar’s television viewership is down 45%, according to an analysis of Nielsen ratings by SportsBusiness Daily, a trade publication. That is twice as large as the National Basketball Association’s decline from its peak, while National Football League viewership has fallen 8%, Nielsen data show.

Tracks have torn out about a fourth of their seats to look fuller but still have wide stretches of empty bleachers on race days. Nascar’s fan base, largely working-class and white, is getting older over all and was hit harder by the recession than the more-affluent fan bases in other major sports.

“There’s no magic pill for this one,” says Ed Rensi, a former Nascar racing-team owner who was a longtime head of McDonald’s Corp.’s U.S. operations. “It’s about economics and demographics.”

Many people in the sport increasingly blame the France family, which runs Nascar and controls racetrack company International Speedway Corp. Long adored for turning fender-crunching races between moonshiners into the nation’s richest and most popular form of motor sports, the founding family’s leadership is now being criticized by drivers and team owners, who fear the Frances are incapable of reversing the fade in fan interest and retreat by sponsors.

Nascar’s chief executive is Brian France, and his older sister, Lesa France Kennedy, is CEO of International Speedway.

One of the most daunting problems is how the siblings’ power is divided, which causes tensions and makes it harder to implement far-reaching changes, according to people throughout the industry.

Richard Petty, a team owner who was so dominant as a Nascar driver that he is considered the sport’s Michael Jordan, complained last summer that owners don’t know who is in charge.

A spokesman says Mr. Petty can tell Mr. France is trying hard but wishes he was more visible at races, like Mr. France’s father and grandfather were.

Brian and Lesa say their disagreements don’t hurt the sport. They say their relationship has never been better, adding that they speak almost every day and are optimistic about stock-car racing’s future.

“We have very strong personalities and express our opinion, but when we get together, we say: ‘What’s best for the industry over all?’” says Ms. Kennedy, 55 years old.

Mr. France, 54, says the downturn reflects challenges faced by all sports as fans increasingly consume content on mobile devices and ticket sales are squeezed by growing demands on people’s time.

He says Nascar also has suffered from a dearth of stars. Dale Earnhardt Jr. was sidelined last year by a concussion but plans to return in the season-opening Daytona 500. “Would we like to be the only one in sports with no headwind? Of course,” says Mr. France. “But that’s how it goes.”

Nascar was born in 1947 in a smoky hotel bar in Daytona Beach, Fla. Bill France Sr., the grandfather of Lesa and Brian, was a gas-station owner who took over a ragtag group of race promoters and created a rule-making body to preside over races.

His son, Bill France Jr., catapulted stock-car racing to prominence by cultivating memorable rivalries between drivers like Dale Earnhardt, who was nicknamed the “Intimidator” for his aggressiveness and died in a 2001 crash, and Jeff Gordon, known as the “Rainbow Warrior” because of the colors painted on his car.

Bill Jr.’s love of hot dogs made just about anywhere and Stag’s Leap Wine Cellars’ Artemis Cabernet Sauvignon from Napa, Calif., encapsulated the sport’s rags-to-riches swing.

After Mr. France was diagnosed with cancer in 1999, he divided his 50% ownership stake in Nascar between his two children, Lesa and Brian. Their uncle, Jim, owned the remaining 50% stake, according to three family advisers familiar with the ownership structure. Bill Jr. died in 2007.

While growing up, the siblings were groomed by their father to take over different parts of the family business. Lesa, a Duke University graduate, began selling tickets at Daytona International Speedway as a 12-year-old. Brian left the University of Central Florida in Orlando after a year and was more interested in competition. He rose from painting walls around Daytona to promoting races at a family-owned track in Tucson, Ariz.

They developed different management styles and ideas about how to advance the sport. Ms. Kennedy says she took after her grandmother, a conservative woman known for managing Nascar’s finances during the early years of the sport, which included making sure the bills were paid.

Ms. Kennedy became interested in refashioning tracks to offer fans views of teams working on cars and building luxury suites to attract a wealthier clientele. Roger Penske, a race-team owner, says the typical Nascar fan makes $35,000 to $45,000 a year. Nascar says average household income of fans is $70,000, close to the U.S. average, citing data from Nielsen Scarborough.

Like his father, Bill Jr., Mr. France pushed for ambitious changes, such as consolidating TV rights from racetracks and selling them in season-long packages, which he succeeded in doing in 2001. That has helped Nascar secure more than $13.5 billion in TV revenue through 2024.

Lesa and Brian worked together to expand the sport beyond the South. Mr. France opened Nascar offices in New York and Los Angeles between 1996 and 2000 and tried to make stock-car racing more like the blue-chip NBA and NFL, says Paul Brooks, a former Nascar senior vice president.

International Speedway, led by Ms. Kennedy, built new racetracks in Kansas City, Kan., and Joliet, Ill., near Chicago. Mr. France overhauled Nascar’s schedule and shifted races away from historic tracks like Darlington, S.C., (nicknamed “The Track Too Tough to Tame”) to newer ones.

Some die-hard fans were turned off by the changes. At Texas Motor Speedway in Fort Worth, Sam Cobb, 41, and his wife, Lisa, 48, reminisce about the raucous parties with stripper poles and kegs that used to be held at a campground near the track. These days, the campground gets quiet at about 10 p.m. on the Saturday nights before big races.

“They’re strangling the fun out of Nascar,” says Mr. Cobb, who misses counting on race weekend for the “largest concentration of rednecks in sport.”

Mr. France and Ms. Kennedy typically don’t spend holidays together and often communicate through emissaries when wrestling with touchy subjects such as scheduling major races, according to a half dozen current and former Nascar industry executives who have worked closely with the France family.

The siblings won’t disclose their exact ownership stakes in Nascar. Four people familiar with the matter say Mr. France sold his entire stake in the company more than a decade ago. He says he still holds equity in the family-owned company.

As a result, these people say, Mr. France essentially works for his sister and uncle even though he is Nascar’s chief executive. That means he runs the sport on a day-to-day basis but is supposed to seek approval from Ms. Kennedy and their uncle for major changes.

She didn’t know ahead of time that Brian planned to announce in 2015 a ban on flying the Confederate flag at races. The announcement came right before Daytona International Speedway, owned by International Speedway, which she runs, was about to host a race.

The company had to scramble to develop a policy on what to do if fans brought a Confederate flag anyway. They were offered an American flag.

Last year, Mr. France endorsed Donald Trump for president at a political rally after being called onstage by the Republican candidate. One racing-industry executive says Ms. Kennedy found out about the endorsement on the news. She donated to former Florida Gov. Jeb Bush’s presidential campaign.

Ms. Kennedy says she can’t recall how she learned her brother was publicly supporting Mr. Trump, adding that it was Mr. France’s personal choice.

Mr. France says he didn’t plan the endorsement until Mr. Trump urged him to speak, adding it didn’t occur to him that supporting Mr. Trump might estrange some of the Hispanics Nascar is trying to lure as new fans.

“I didn’t calculate it that way,” says Mr. France. “Maybe I should have.” Mr. Trump drew 29% of Hispanic voters on Election Day, according to exit polls.

Three-time Nascar champion Tony Stewart said last year in a radio interview that Mr. France should pay more attention to the sport and attend more races.

Mr. France says he went to roughly half of the race weekends last season. He says Nascar teams, drivers and auto makers are working more closely than ever to improve competition and boost interest in the sport.

Most of the 13 tracks owned by International Speedway rely on hosting two top-tier races a year for the bulk of their ticket revenue. Yet Nascar makes the race schedule, with an emphasis on attracting the most possible TV viewers.

The conflicting agendas were evident in recent discussions with broadcast network NBC, which pays Nascar about $440 million a year. Sixty-five percent of the total is steered to tracks, 25% to teams and 10% to Nascar.

Executives at NBC, part of Comcast Corp.’s NBCUniversal unit, raised the possibility of moving some Nascar races to the middle of the week, says Jon Miller, president of programming at NBC Sports.

That would limit the number of Sunday races that compete with NFL games for viewers, possibly boosting TV ratings but hurting attendance in person. Race fans often travel a long way and attend pre-race events that can go on for several days or longer.

The idea hasn’t been discussed since the fall and isn’t a priority right now, says Mr. Miller.

Nascar did reach a five-year agreement with racetracks that makes it easier for them to plan big spending projects. The downside is even less flexibility for Nascar to dramatically alter its schedule and take races to new places.

Ms. Kennedy says she is doing whatever she can to recapture longtime fans and cultivate new ones at her racetracks. She oversaw a $400 million renovation at Daytona, completed last year, that added escalators and cushioned, leather seats often seen at NFL stadiums. She wants to install Wi-Fi at tracks and add luxury suites or clubs that attract more affluent ticket buyers.

“The next generation is looking for more rapid entertainment, more interactive entertainment,” she says. “They have so many options available that you have to have a compelling story.”

In October, the siblings went to NBC Sports headquarters in Stamford, Conn. TV ratings for the Nascar season were headed for another decline. NBC executives pressed Ms. Kennedy and Mr. France to make radical changes.

They agreed to take action. In December, Nascar gathered racing-team executives, drivers, track operators and TV executives at the Wynn Las Vegas hotel. The siblings didn’t attend.

“Everyone loves the same thing, but you have different opinions on how to get there,” says driver Brad Keselowski, who won the Nascar championship in 2012 and has openly criticized the Frances. He compares the Las Vegas meeting to couples counseling. “Everyone has their baggage, and you work through it and try to give it a shot,” he says.

The group and Nascar decided to divide each race this season into three stages, awarding points to the top drivers at the end of each stage. The Daytona 500 will be the first major race under the new rules, and Nascar is working on at least a dozen more potential changes.