Perhaps it would help with recommendations if you told us what American shows you like, or for that matter, which you’ve liked of the British TV shows you’ve seen. (I can’t tell if your list includes shows you’ve already watched or if those comprise your ‘yet to be seen’ list.) Do you just want comedy? Is drama okay? Do you object to old series? That sorta thing.
My own recommendations not already, uh, recommended, would include:
Red Dwarf - if you’re a sci-fi fan, but not one who demands adherence to any continuity (or even any realistic premise), this is a must-see. On the mining ship Red Dwarf, the only survivors after a terrible accident are slobby, lazy, but good-hearted Dave Lister, the only human left alive three million years after the death of his crew (he survives thanks to having been in statis as punishment for bringing a cat on board the ship); the Cat, a vain, selfish, not-very-bright sentient being who evolved from the aforementioned cat; Holly, the ship’s senile A.I./computer; and Arnold Rimmer, the hologram projection of Lister’s old nemesis/bunkmate, whom Holly revived because Holly believed that opposing Rimmer’s personality – a mass of neuroses, cowardice, thwarted ambition, and self-loathing – is probably Lister’s best shot at remaining sane and productive. (Rimmer’s the best thing in the series by a long shot.) Certainly silly but also imaginative, incisive, and quite surprisingly poignant at times.
House of Cards - if you like Yes Minister, here’s another show (really a mini-series) about the machinations at the high levels of UK politics. Ian Richardson plays Francis Urquhart, the Chief Whip who’ll do [del]almost[/del] anything to crawl his way to the top. Makes Sir Humphrey look like a fluffy bunny. Funny, cruel, devastating. There are two sequels as well, so all in all you’ve got about nine hours to enjoy IIRC.
The Good Life, aka Good Neighbors here in the States. I recently mentioned this show in a different thread; it’s a highly underrated show – at least, in the U.S. it is; in the UK it’s a perennial favorite, wise folks that they are – about a couple of suburbanites who decide to leave the rat race and take up self-sufficiency, from making their goat’s cheese (courtesy of their own goat) to making their own gas generator from pig/chicken/goat manure … all while still living in their Surbiton upper middle class neighborhood. Much to the consternation of their next-door neighbors/best friends, who alternate between irritation and support of the Goods’ unorthodox ways and the many effects their lifestyle has on their friendship. Perhaps the best ensemble cast in sitcom history, tied with the trio from Yes Minister and the quad from All in the Family. The humor here is gentler than the other two shows, and tends to be much more observational takes on society/relationships. Lots of witty fun and the performances are just stellar. Also, I think the relationship between Tom and Barbara Good is the most delightful depiction of a married couple on TV.
Without knowing your taste I’ll leave this for now. 
Though I’ll add, if you like relationship/sex humor, you might enjoy Coupling, which apart from some awfully stiff performances from pretty much the entire cast bar one (Richard Coyle’s Jeff), grows on you quite a lot.