NAYMIYAH, Iraq — A column of black Humvees carrying Iraqi special forces rolled into southern Fallujah on Wednesday, the first time in more than two years that government troops have entered the western city held by the Islamic State group.

The counterterrorism troops fought house-to-house battles with the militants in the Shuhada neighborhood, and the operation to retake the city is expected to be one of the hardest yet.

“Daesh are concentrating all their forces in this direction,” said Gen. Haider Fadel, using the Arabic acronym for the Islamic State militants.

Prime Minister Haider al-Abadi promised a swift victory when he announced the start of the operation on May 22 to liberate Fallujah, 40 miles west of Baghdad. But the complexity of the task quickly became apparent.

Although other security forces from the federal and provincial police, government-sanctioned Shiite militias and the Iraqi military have surrounded the city, only the elite counterterrorism troops are fighting inside Fallujah at this stage of the operation. And they are doing so under the cover of U.S.-led coalition airpower.

“We expect to face more resistance, especially because we are the only forces entering the city,” Fadel said.

The Islamic State has suffered setbacks on several fronts in the region where it captured large swaths of territory two years ago. In northern Syria, U.S.-backed rebels made a final push Wednesday in the town of Manbij – a waypoint on the Islamic State supply line to the Turkish border and its self-styled capital of Raqqa. And in Libya, forces loyal to a U.N.-brokered government have advanced inside the coastal city of Sirte, the stronghold of the Islamic State’s local affiliate.

Fallujah is one of the last Islamic State strongholds in Iraq. Government forces have slowly won back territory, although the militants still control parts of the north and west, as well as the second-largest city of Mosul.

It is symbolically important to both sides: Many of the Iraqi forces fought al-Qaida – the predecessor to the Islamic State – in this same territory, and the city was the scene of some of the bloodiest urban combat with U.S. forces in 2004.

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