I think American comedy is catching up, but it depends on what you like, really. I love shows like Brooklyn Nine-Nine, (parts of) How I Met Your Mother, and a lot of the output of Adult Swim. But I really love dark, awkward and/or surreal comedies, like It's always sunnyin Philadelphia, Veep (though, does this count really considering who made it) or Portlandia.
I think there is just more of this sort of recent(ish) output in the UK to appeal to me- Black Books, Nathan Barley, Brass Eye, The Thick of it, Fresh Meat, Mighty Boosh, IT Crowd, Inbetweeners, Psychoville, League of Gentlemen, Father Ted, Extras... I think the fact that there is less censoring on British tv allow creators to go a lot further for a gag (see League of Gentlemen), whereas some comedies in the US are limited in what they can do (way more that American dramas, which are superb). In terms of really mainstream shows, I think Big Bang Theory, Two Broke Girls and that ilk are just as bad as similar shows here. But Britain does great cult comedy, and it does it better than most.