Conan O’Brien and his family were out to dinner in Santa Monica last year when his daughter began to screech, “Oh my God! Oh my God! Oh my God!”

“I thought a Cessna had just plowed into the sidewalk and burst into flames,” the late-night TV host recalls. “Then my son started to freak out and he was like, ‘They’re crossing the street! They’re crossing the street!’”

The source of the pandemonium was the arrival of what Mr. O’Brien’s children deemed some bigger celebrities: a few mild-mannered Mormons. The late-night TV host, who soon took a picture with them, recognized them as the stars of Studio C, a sketch comedy show out of Brigham Young University.

Studio C has achieved sizable popularity on the internet, despite—or perhaps because of—its super-scrubbed brand of clean humor, such as a skit about a soccer goalie named Scott Sterling who accidentally, and agonizingly, blocks shots with his face.

Working blue is out of the question for this comedy troupe. BYU, run by the Church of Jesus Christ of Latter-day Saints, has ranked as the nation’s most “Stone Cold Sober” place of higher learning for 20 straight years, according to the Princeton Review.

The performers are employees of the Provo, Utah, school and must adhere to its honor code, which includes following church principles such as eschewing smoking and avoiding coffee. Male cast members must obtain university permission to grow facial hair—dubbed a “beard card” by students.

Writers at Studio C, which launched in 2012 and began its new season this month, must avoid innuendo, cursing, politics—even the word “gosh,” because it sounds too much like “God.” Flatulence jokes don’t stand a prayer of getting past the BYU television censors. But bits like the appearance of curmudgeonly Harry Potter character Severus Snape on a “Bachelorette”-like reality show—that works.

A skit where Harry Potter character Severus Snape appears on a reality show. Studio C

Studio C video where Harry Potter character Severus Snape appears on a ‘Bachelorette’-like reality show.

The result: a burgeoning pop-culture phenomenon that has racked up more than 1 billion views on YouTube—about a third of the number of “Saturday Night Live.” Studio C’s success, flying in the face of comedy’s subversive ethos, is an irony not lost on university officials.

“Here we have arguably the most successful university-based comedy troupe of all time coming from what the world would consider to be a conservative religious school,” says Michael Dunn, managing director of BYU Broadcasting. Studio C’s popularity has validated the idea that “the absolute sharpest comedy is clean comedy.”

Still, some have found reasons to be offended. One viewer wrote in to complain about a joke at the expense of a character with a hernia, saying that “hernias are painful.” Others objected to a bit where people shot at a cat. Another chastised a cast member for using the word “butt” in a social media post, suggesting it would have been better to opt for “derrière.”

Matt Meese, left, in a Studio C parody of ‘Who Wants to Be a Millionaire.’

Matt Meese, left, in a parody of ‘Who Wants to Be a Millionaire.’ Studio C

To find laughs, Studio C uses familiar comedic archetypes, such as the “fish out of water,” exaggeration and pop-culture parodies, says Matt Meese, 34 years old, the show’s co-creator and one of its 13 cast members. They try to raise the stakes for every situation as far as they can go, he says. A unifying theme of many sketches is “struggle.”

Often, such as in the Scott Sterling sketch, the struggle involves physical pain for Mr. Meese, or seemingly being the only sane person in the scene. On a “Who Wants to Be a Millionaire” parody, his character is thrown off when the entire studio audience, his wife and a Stanford University historian all insist that the 26th U.S. president was George Washington. Against his better judgment, he selects Washington as his final answer, and loses $1 million.

In “Two Guys on a Scooter,” a trip to the grocery store between friends takes a turn when 11 people, including a grandfather and a police officer—all carrying items like a bicycle or a watermelon—pile onto the tiny scooter.

Studio C’s “The Crayon Song Gets Ruined” video.

Studio Cs ‘The Crayon Song Gets Ruined.’ Studio C

Even without swearing or references to sex, Studio C can tap into comedy’s subversive roots. In “The Crayon Song Gets Ruined,” a simple, happy tune sung by crayons pointing out that roses are red and pumpkins are orange is upended by a depressed cohort. He continually interrupts the song, linking red to blood, orange to prescription bottles and yellow to jaundice.

Mr. O’Brien, a former writer for “Saturday Night Live” who now hosts his own late-night show on TBS, says the cleanliness of Studio C’s humor was almost an “afterthought” to him. What got his attention was the craftsmanship of the skits, particularly their solid endings, something he has always found challenging.

“If you’ve ever seen a church group doing a sketch, you’re very aware the whole time that this has been sanitized,” he says. “But for them, it didn’t hurt the comedy at all.”

Cast members, who are current or former BYU students, receive regular paychecks and benefits. They say they know intuitively what subjects are taboo. But on occasion, they see what they can get away with.

Despite BYU’s aversion to potty humor, they have tried to get flatulence into several sketches. One attempt involved a James Bond-style torture scene where the hero turns the tables on his captors by passing gas. Although the skit never mentioned the word “fart” and featured no expulsive noises, a BYU broadcasting official rejected it.

“It was so classy,” says Mallory Everton, one of the cast members. “It was just actors exchanging looks. We went full highbrow on a really lowbrow idea.”

As Studio C’s popularity has grown, cast members are routinely besieged by fans, who have asked for selfies at the Eiffel Tower, Disneyland, church and even at a funeral. In Provo, some of the cast members have attempted to go in disguise.

The comedians, many of whom are former missionaries, said they don’t mind getting the Hollywood treatment and like appealing to a wide audience. Last year, an apostle, one of the most senior leaders in Mormonism, came to a taping.

Cast member Jason Gray said the experience was intimidating, like doing sketch comedy in front of the pope. Still, he performed an impression of the church official. “It was surreal,” Mr. Gray said. “It’s comedy so clean that literally an apostle can go and have a good time.”

—Erich Schwartzel contributed to this article.