“The enemy has made a decision to make Kobani his main effort,” General Austin said on Friday.

The United States responded by sharply escalating the bombing campaign in conjunction with Kurdish fighters, who supplied targeting coordinates. Though American officials denied that the situation in Kobani was a factor, the stepped-up airstrikes began as the siege became a subject of international media scrutiny: Because of an accident of geography, journalists, refugees and others were able to watch every turn in the battle from hills across the border.

“Now, my goal is to defeat and ultimately destroy ISIL,” General Austin said. “And if he continues to present us with major targets, as he has done in the Kobani area, then clearly, we’ll service those targets.”

As the militants have been put on the defensive over the last few days, the number of airstrikes has decreased. American military officials said they had conducted six airstrikes in Kobani on Thursday and Friday, a sharp decline from the 37 attacks carried out over a three-day period earlier in the week.

“ISIS is retreating and we are advancing,” said a commander serving with a unit of Syrian antigovernment rebels who have allied themselves with the Kurds.

“Airstrikes destroyed most of their heavy weapons,” said the commander, who uses the nom de guerre Abu Hasan. The militants had also lost their supply lines and were no longer able to move freely in and out of the city. “Those who are left in the city center don’t have a way out,” he added.