Peering from his trench on a ridgeline fortified with sandbags, the Kurdish commander gazed toward an Islamic State-held village next to territory recently seized from the extremist forces. Smoke wafted into the sky from natural gas burning off a well in a broad basin known as Wadi al Naft, or Valley of Oil, west of Kirkuk, Iraq’s strategic northern energy hub.

“Daesh is weaker now, no doubt,” said Hussein Yazdanpana, who heads an Iranian-Kurdish front-line unit, using an Arabic acronym for Islamic State. “We’ve heard that many of their fighters are running away,” added the militiaman, who incongruously wore an American flag pocket patch — a gift from “Jack,” one of a number of a U.S. military advisers here.

While media attention has focused on Iraqi government advances north from Baghdad on the Islamic State-held city of Tikrit, the Sunni militants have also been suffering a series of significant reversals in the Kurdish north. […]