Here is a map of showing the areas in the region with Kurdish majorities, and the territorial claims made by Kurdish groups at various points in the last century. As you can see, a Kurdish state could potentially include a very large chunk of southeastern Turkey (note espeically the light purple line, the frontier claimed by Kurdish nationalists in 1945). The Turks have fought a bitter campaign against Kurdish separatists, with lots of ham-handed nastiness on the Turkish side: until recently, it was illegal to use the Kurdish language in Turkey, and the Turks even denied there were any such thing as “Kurds” in Turkey–the officially preferred designation was “mountain Turks”, which is historically and linguistically absurd. Of course, the Turks for their side could point to assorted atrocities carried out by Kurdish guerrillas. At any rate, the Turks fear that an independent Kurdish state in northern Iraq, far from satisfying Kurdish national aspirations, would simply serve as a springboard to launch campaigns for a united Kurdistan in all Kurdish-inhabited territory, which could claim areas far beyond the relatively small enclave currently controlled by the Kurds in northern Iraq, and even beyond the larger area of Iraq in which Kurds are a majority of the population.
I don’t think the U.S. is in any way supporting an independent Kurdistan; I’m sure our policy would be to accomodate the national aspirations of Iraqi Kurds within the present borders of Iraq, through some sort of federalism. And I believe the current leadership of the autonomous Kurdish enclave in northern Iraq has been very careful to take a reasonable tack–autonomy for Iraqi Kurds within a federal Iraq, not total independence and unity for all Kurds, which would embroil the Kurds, Turkey, Iraq, Iran, and probably Syria in a huge regional war. But there’s no question the Turks fear any greater recognition of Kurdish national apirations would be the thin end of the wedge. (It’s also worth pointing out that despite the strength of Kurdish nationalism the Kurds are not internally united; within Iraq, there are two main Kurdish nationalist movements–I believe they still have the Kurdish autonomous enclave divided into two essentially separate states, one under the control of each party, although I think there is freedom of movement between the two sectors and generally peaceful relations between them–and Iraqi Kurdish groups don’t necessarily get along with or trust Kurdish groups from Turkey.)