I believe they are different than Saudi/Kuwaiti types (Iraqi women don’t have to wear bags on their heads to go outside), but as far as their love for us - forget it. It was literally distrust of our Iraqi allies that forced the midnight retreat. The pictures from the embassy in Saigon n 1975 had become such symbols of failure we bugged out in Iraq when no such pictures could be taken.
You’re right; Gulf Arabs are more fanatical in oppressing women, both in terms of more extreme veiling like niqab, abaaya, and burqa, besides work and other social restrictions. However, Islamization, including hijabization, has advanced in Iraq, as well. This was utterly predictable. Remove a relatively secular tyranny in a devoutly Muslim country, and Islamaniacs are sure to fill the gap. Politics, like nature, abhors a vacuum.