RAMADI, Iraq—Amid the rubble of hundreds of airstrikes that helped drive Islamic State fighters from the center of this city, almost the only people in sight earlier this week were among 500 members of an elite unit of U.S.-trained forces, credited by Iraqi officials with carrying out the bulk of the fighting.

On Friday, nearly two weeks after the government declared that it had retaken control of Ramadi, members of this unit had only just subdued the eastern neighborhood of Mala’ab with little help from the regular army,...