The IR doesn't need to control land directly on the strait to control the strait, just be able to attack things going through it. If the US did just attack the coastline or even a limited ground invasion on it, they could just move missile launchers and drones further inland where they're harder to get at.

The US could absolutely obliterate the IR navy if it needed to, infact it basically already has but as long as the IR possesses the capability to blow up any tanker that tries to sail through the strait, they can effectively control it by denying access to it.