The fights are as real as the bikes on Discovery Channel's surprise hit American Chopper, its stars tell Denis Staunton in upstate New York.

It's an hour of yelling, door-slamming and motorcycle-building that can literally mean watching paint dry, but American Chopper has become the Discovery Channel's biggest hit and one of the world's most successful reality TV shows.

The show has turned Paul Teutul and his sons, Paul jnr and Mikey, into international cult figures and has transformed Orange County Choppers, within three years, from a tiny customised-bike workshop employing three people into a global brand with a full-time staff of 70.

The Teutuls have built bikes for actors Bill Murray and Will Smith, cyclist Lance Armstrong and chat-show host Jay Leno, and Paul snr says the celebrity orders just keep rolling in.

"We don't have a shortage of people coming to us. We're talking big celebrities, and they're all knocking at the door," he says.

A massive, muscle-bound bear of a man with a walrus moustache, Paul snr acknowledges, however, that American Chopper's appeal lies less in the motorcycles than in his relationship with his sons, particularly with Paul jnr, the laid-back bike designer and creative force behind the business.

Paul snr insists that the father-son arguments, usually over Paul jnr's slow pace and poor timekeeping as a deadline looms, are all unplanned and televised exactly as they happen.

"Most reality shows that I know of are scripted, and if they're not scripted, they're done so that, if they're doing a scene naturally and if it doesn't look right, they re-do it," he says. "Here, that never happens. If we're having a fight, we're having a fight. We're not going back and start fighting again. If we have a conversation, we're not going back to have it again. That's the way it is. And it's pretty obvious that that's the way it is."

A six-person crew films constantly at the Orange County Choppers workshop in Montgomery, in upstate New York, from 7am to 6pm six days a week. The Teutuls say they no longer notice the cameras but Paul snr says the shows, which often involve emotional or even tearful exchanges, can make uncomfortable viewing for him.

"It's quite embarrassing," he says. "The problem is that we forget that the cameras are there, and it doesn't start out that we're going to get emotional. We're discussing something about things that he doesn't like. And I'll tell him about things that I don't like, and wherever it winds up, it winds up. You don't say, oh shit, I'm on camera. It's after the fight's over that you have time to think about it. But it's too late."

If he could change anything in the family dynamic, Paul jnr says he would like his father to be more emotionally stable and to acknowledge that the creative process can be unpredictable and frustrating. He claims to dislike arguing, suggesting that, if he and his father got on better, they would add 10 years to their lives.

"We didn't set out to be in the public eye as a family that argues," he says. "That was not our intention. And it's really not our intention on a daily basis to do that. It's just the way it is."

The format of American Chopper is unchanging: a commission to build a customised bike, Paul jnr's agonising over the design, and his father's explosions as the deadline approaches. Younger son Mikey is the family clown, a fun-loving slacker who does little work but can do no wrong in his father's eyes.

"Mikey's contribution to the business is his personality," his dad says. "So it's pretty apparent that that's part of the glue that keeps everybody together. If everybody agreed and everything ran smoothly, it would be boring for me and I think it would be boring for the viewers also."

The Teutuls' Montgomery headquarters includes a fully equipped gym, a seldom-used basketball court and a warren of offices to manage the expanding business. They are building a massive new complex nearby, with a showroom, a multimedia exhibition area and a visitors' centre that will allow fans to watch the family at work through a one-way mirror.

Later this year, the family will launch a cartoon show, to be broadcast on Saturday mornings. New clothing lines are also on the way and a distribution deal has been signed with Triumph motorcycles.

"We make a lot of money on our merchandise and it's a real, intricate part of our business which came with the fame," says Paul snr. "But the bike-building is our core activity and that's what we constantly pay attention to. We have people in areas paying attention to our merchandise, but it's not our world. Bike-building is our world."

With audiences growing each year, however, American Chopper looks set to run for a long time to come and Paul snr believes the constant arguing strikes a chord with families the world over.

"I think in most families there's a lot of interaction like that, whether it's in business or at home," he says. "So I think a lot of people identify with that. A lot of families are dysfunctional."

• American Chopper is on Discovery Channel on Sundays at 8pm and Mondays at 9pm.