An American women getting married to an Iraqi man how ?

This is a very complicated situation.  You have not given much details i.e. are you resident in Bahrain?  is your fiance to be resident in Bahrain?  is he muslim?

If either of you are not resident here; you may manage to end up getting married in a place of worship (depending on religious belief) with letters from your embassies but getting the certificates from Ministry of Justice in Bahrain may be a problem. 

You should be able to register the marriage in your embassies though.  See the US Embassy link below:

https://bh.usembassy.gov/u-s-citizen-se … /marriage/

Also this thread on the forum is addressing the general topic of marriage in Bahrain for expats (might not be fully relevant for you but can potentially clarify some elements for you):

https://www.expat.com/forum/viewtopic.php?id=150315

As for getting your fiance over; he has to apply for a visit visa, either through a travel/tour operator or directly on the evisa site if he is eligible.  You can't sponsor him as both of you are not married yet.  Word of caution though; visas for certain nationalities are not easy to get and his nationality is one of them.