Is British TV Crime Drama Better than U.S. TV Crime Drama?

Probably.
Throw me any good U.S. show and I'll put up a better U.K. offering (or original). The  U.K. shows I am prattling on about are all free-to-air television (as opposed to HBO and other U.S. cable networks).
Those C.S.I. <insert interesting U.S. city here> series are running out of puff.
You get more bang for your buck with British crime shows, particularly featuring psychos. Take a recent episode of BBC's "Silent Witness" (it started a decade before C.S.I Crapola started). You had a serial killer that made Joe Carroll from "The Following" look like Mr Rogers. All in two (1 hour each) episodes.
Take ITV's "Broadchurch" for superb character development. "Wire in the Blood" for tormented, gifted profilers, long before Will Graham in "Hannibal". "Waking the Dead" also blitzes its U.S. counterparts.
Then you have three more ingredients to add more texture - the British class system, historical locations and eccentric (nutbags?) characters. Take "Midsommer Murders" (please, some wags would say).
Take the Brit's heritage of murder whodunits - Agatha Christie, Sir Arthur Conan Doyle, G.K.Chesterton, Ruth Rendell). The Brits do 'gritty' as well as graphic. Take old episodes of "The Bill" - even minor characters were so well cast, the acting so polished. I remember the high quality of 1960's shows like "Z Cars" and "Softly, Softly".

Don't get me wrong, there are a few standout U.S. stars of the genre: I'm sure you can think of others...like "The Sopranos" and "The Wire" (both cable, though).
"Hannibal" is remarkably brave and multi-layered for free-to-air television. "Dexter" (HBO) was groundbreaking, hope the last season goes out with a bang.
Then to add confusion, we have U.K./U.S. co-productions like the delightful "Ripper Street".