The democratic majority in Iraq is Shi-ite Muslim. They're aligned with Iran. (That 100% predictable outcome was ignored by the Bush administration when it invaded in 2003.) So given that the USA and its Gulf allies are deeply hostile to Iran, how will they counter Iran's influence over the government of Iraq?

By tacitly supporting Sunni Muslim extremism in Iraq, the USA and its allies have a powerful counter to the influence Iran has with the Shi-ite Muslim majority. Hence, the fall of Islamic State will not bring the end of that extremism. It's just too useful a tool for the US to discard. Islamic State may not continue, but Sunni Muslim extremism in the region will.