At the time of Gulf War II under W, the population was about 20% Sunni, 20% Kurdish, and 60% Shiite. So if you disarm the minority ruling Sunni establishment and instead pledge to install a democracy, a Shiite government is the logical conclusion.
Since realizing their mistake, the US has attempted to patch together some multi-religious, multi-ethnic government, but these are all destined to failure. The Kurds want to be left to their own devices, the majority Shiites want to rule the south of the country with most of the oil, and they want the Sunnis left out of power in the deserts to the West.
George HW, being smarter than George W, pulled his punches, left Saddam in power, and left him with helicopters to kill enough Shiites to prevent a successful rebellion of the "marsh Arabs", as they were called in news reports back then.