Dominates in what field.
To break it down on the political, economical and military level.
Militarily the ISF which is compromised out of the MoI's police forces, MoD's military, ISOF and the PMU. ISF is dominated by forces that are working closely with the thousands of US troops on Iraqi bases. Arms imports are mostly from Russia, US and China.
Economically it may be right for certain parts of Iraq, the south is definitely importing a lot from Iran. Though this is simply due to the available options with regards to the geography. The only other industrial nation that produces a lot of goods is Turkey and Iraq imports a lot from them as well. The right answer to this is local production.
Politically Iran is not dominating either. Some describe the current situation as Abadi (pro US camp) and Maliki (pro Iran camp) with the latter having a lot of influence in the PMU and the first having influence in the other military forces. No side is fully dominating, there is a certain balance of power.
http://abcnews.go.com/International...n-check-isis-fight-official/story?id=32651016
In 2014, Iran made an offer to Baghdad government. Iran offered full use of its military equipment and troops, that means Iran offered to dispatch a massive number of its military to fight IS in Iraq. Iraqi gov rejected this request, instead invited the US-led coalition to Iraqi bases and the airspace.
Explain the logic of that if Iran was in control?
It's this decentralized power system on top of which there is a balance of power between US/Iran leaning actors which lays the fertile ground for this shitty corrupt weak state system. If Iran or any other side was fully in control of Iraq the country would be stable, the 2014 collapsed would never have happened. We'd have seen a Saddam style strong state, the force in control is irrelevant to that.
Instead we have internal enemies (Kurds) leeching from the system sabotaging the state on every field they can, fortunately they're being kicked out of positions but it's not enough. Nationalists have to return to power, all these religious figures need to be kicked out.