SALHIYAH, Iraq — Kurdish fighters in northern Iraq captured a border crossing with Syria on Tuesday, expelling Islamic State militants in heavy fighting that ground down to vicious house-to-house combat and close quarters sniping.

In neighboring Syria, Kurdish militiamen were on the defensive as the extremists pressed ahead with a relentless assault on a town near the Turkish border. The attack on Kobani, also known as Ayn Arab, has driven more than 160,000 people across the frontier in the past few days.

Iraqi Kurdish fighters, known as peshmerga, were doing the bulk of the fighting on the ground as a U.S.-led coalition carried out an aerial assault against the Islamic State group in both Iraq and Syria. Britain joined the air campaign Tuesday, carrying out its first strikes against the extremists in Iraq — though it does not plan to expand into Syria.

The goal of the campaign is to push back the militant group that has declared a self-styled caliphate, or Islamic state, ruled by its brutal interpretation of Islam in territory it has seized across much of Iraq and Syria.