Whatever they think of modern-day conveniences such as electricity and power steering (hard with a horse and buggy) the American Amish might want to consider the services of a public relations firm. Someone needs to counter the awful stories appearing in recent weeks that have featured rape, incest and the gassing of dogs, lots of them.

The landslide of bad news – "A Crisis in Amish Country" was a New York Times headline this month – coincides with something else unexpected about members of a sect descended from Anabaptists who fled Europe to the US, mostly to Pennsylvania, in the early 18th century. In population terms, at least, the Amish are booming. And as their numbers swell – growing at a rate of 5 per cent a year – they are also beginning to migrate westwards.

According to a new study, the Amish, many of whom still eschew English and speak Pennsylvania Dutch (actually a German dialect), are quickly moving beyond their traditional rural bases in Pennsylvania, Ohio and Indiana. The survey, completed by researchers at Elizabethtown College in Pennsylvania, says they are now settled in significant numbers in 28 US states. There are now just over 250,000 American Amish altogether and, at current growth rates, the population may double by 2024 if not before.

As they grow – one main reason is the tradition of having five or more children per couple – the Amish may struggle more than ever to manage relations with the outside world, which regards them with a mixture of condescending curiousness – Amish farmers' markets are a big draw even in Manhattan – and outright suspicion. The Amish have their special exemptions: they don't pay some federal taxes and they educate their children in their own schools.

That they can skirt the rules does not sit well with everyone. Last week's news about an Amish dog breeder in upstate New York will not have helped. Officials accused David Yoder, of Black Diamond Kennels in Romulus, of connecting a hose to an engine (apparently combustion power is allowed on his premises) and then to a small shed filled with dogs. He allegedly killed 93 dogs in his makeshift "gas chamber" after refusing to heed a Department of Agriculture order that he test and vaccinate the animals against brucellosis.

More damaging, however, has been the case of Chester Mast, a 26-year-old Amish man who last week pleaded guilty to charges of rape, sodomy and misconduct with a child in Missouri. Married with two children, Mast was sentenced to 14 years in prison and still faces related incest charges in neighbouring Wisconsin.

That case was especially shattering for the community of Amish 75 miles north-west of St Louis where he was living. The most traditional Amish orders usually strive to handle all crimes committed by members of the fold themselves, often with corporal punishment or excommunication, temporary or permanent, from the church. In this way, they also protect their own from prosecution by the mainstream judiciary. In the case of Mast, however, the Amish leaders turned him over to the police.

"Some Amish communities aren't fully aware that a psychological disorder may be underlying devious behaviour," said Professor Donald Kraybill of Elizabethtown College. "They may sometimes confuse this kind of an addiction – like an alcohol addiction or a sexual addiction – with a spiritual or moral weakness. They think that if the person confesses the sin, and they bring them back into the church, and they pray about it, everything is going to be OK."