Here is a brief chronology:
US backed the Kurds in 1972, possibly to destabilize the new government of Iraq under Gen Qasim which was nationalizing Iraq’s oil resources.
The US dropped support for the Kurds in 1975, possibly due to the Shatt Al Arab waterway dispute resolution in which Iran dropped its support for dissidents in Iraq.
The US didn’t come to the aid of the Kurds during the Anfal campaign, instead supported Saddam Hussein with money and weapons technology, during the Iran Iraq War 1980-1988.
The US gave verbal support in 1991 to the Kurds , encouraging them to rise against Iraq, but didn’t follow through. Saddam Hussein attacked the Kurdish revolt, millions of Kurdish refugees were left stranded on mountains on the Turkish border. The US did eventually provide an umbrella of protection (UNSC Res 688)
The US supported Kurdish autonomy leading up to and after the invasion in 2003, but in 2007 ‘turned a blind eye’ to the Turkish incursion into Kurdish territory in Turkey's pursuit of Kurdish militants. Turkey is a US ally.