The mistake was actually letting Iraq enshrine Islam in their constitution.
The purpose of the war was to make it a secular democracy that could be prosperous and would blunt the force of the most aggressive Sunni Islamists (including Saudi Arabia, where most of the hijackers originated). Saddam was doing his own version of a secular but nominally Islamist (Sunni) state, but it’s so difficult - because Islam is also a political system - that I think it had escaped even him. His Baathist party was based on a somewhat syncretist version of Islam and the reason that he was so oppressive (other than the usual dictator’s arrogance) was that neither side wanted to join the party.
The Iranians are mostly Shiites, so we backed off on taking down Mookie in Iraq because it was mistakenly thought that we could benefit from a Sunni-Shiite face off.
And in any case, by then, the Dems (who had suddenly decided that Islam was just the coolest thing going) had totally worn Bush down and this was really his biggest mistake.
The war was fought well and we won in very difficult circumstances. As usual, it was the peace that was lost. We should never have gotten involved in Islamic politics and simply should have imposed a non-confessing state. Also, Muslims only respond to dictators and autocrats, and had no respect for us when we started trying to be subservient to supposed Islamic sensibilities.
We should have adopted the model we used in Germany after WWII.
That wasn't a mistake -- that was a warning flag.
Anyone with half a brain should have seen right then that the whole "war on terror" was a giant fraud that was perpetrated on the American people by a corrupt, diabolical government.