In all honesty, I don't care that much. Iran will probably manipulate them for a while, but they'll likely develop more of an independent mindset over time.
Iraq has a Shia majority that was historically at the bottom, a Suni minority that was historically at the top, and a Kurdish minority (Suni and Shia) who historically want their own country but cannot make it a reality due to all their neighbors inside and outside Iraq. The Shia have a strong connection to Iran and the Suni to the Suni Arabs of the Middle East and the Kurds are usually canny politicians. So in my opinion the balance of power in Iraq rests with the Kurds. If they can stay relatively united (a difficult task since the Kurds are politically and militarily divided) and help offset the Shia majority Iraq may stay relatively peaceful and democratic: if they do not, the Iranian influence will become more and more pronounced and Iraq will become less and less peaceful. The one bright spot is the Islamists are discredited for a while among the Suni because of their arrogant and violent behavior during the Iraq Insurgency and the Shia want little to do with the Islamists.
Why the hell do you care?? Iraq (and Muslims in general) is only dangerous because Jew led American liberals keep interfering in their affairs. Leave Iraq alone. Let them sort their own selves out.
With 65% of Iraq's population, it's only a matter of time before the Shia majority will dominate the political landscape. Expect big trouble if they connect with their Iranian brethren.
Go read the Iraqi constitution. It states that Iraq the law of the land in Iraq is the QUR'AN. I am not sure if that meets your definition of Sharia law, but Iraq IS an Islamic nation under the law of the RELIGIOUS BOOK of the Islamic faith.
Iraq and America will both break up in the long run. Iraq is a fusion of three separate identities. America is a fusion of even more separate identities. We can't get along.
