As the sound of outgoing artillery and heavy machine-gun fire rang out across the undulating fields outside Makhmour, trucks bearing fresh supplies of ammunition and SUVs carrying uniformed officers hurtled to reinforce the front lines.

Hundreds of volunteers drawn from all over the Kurdistan region also streamed toward the battle, clutching ancient rifles and wearing the ballooning pants and waistcoats traditional to Kurdish culture.

The first of the three U.S. airstrikes had taken out an Islamic State artillery position nearby, and pesh merga commanders said they sensed the militants had been chastened by the attack.

“This power they had before, this momentum — we don’t see it now,” said Col. Mohsin Avdal, who sat poring over maps on an ammunition box beside a pile of several dozen newly arrived 107mm rockets. They were delivered, he said, from stocks the pesh merga already owned.