The U.S. didn't invade Iraq as a response to terrorism. They invaded Iraq because of the threat of "Weapons of Mass Destruction". Many people just assumed Iraq was behind 9/11 simply because the government wanted to invade them after 9/11, and the government did little to dispel that mistaken notion. And we all found out that Iraq having "Weapons of Mass Destruction" in the first place was a mistaken notion as well.
Fear of terrorism was one of the reasons for public support of the war and it was heavily pushed in the pro-war propaganda.
> They invaded Iraq because of the threat of "Weapons of Mass Destruction".
In the run-up to war, both supposed direct ties between the Iraqi regime an al-Qaeda (and direct ties between Iraq and one of the 9/11 attack specifically) and the threat that if Iraq acquired WMDs it might transfer them to al-Qaeda or other terrorist groups for use against the US were trumpeted. WMD and terrorism are not mutually exclusive motivations for war.
The supposed connection between the Iraqi regime and al-Qaeda (including a supposed link between Iraq and 9/11 hijacker Mohammed Atta) was still be trumpeted months after the 2003 war began, even just before the 9/11 Commission issued its finding that no such connection existed. [0]
[0] http://www.nbcnews.com/id/5223932/ns/us_news-security/t/pane...