Masrour Barzani, the Kurdistan Regional Government’s intelligence and security chief, described his forces as “overstretched.” In an interview this week, he called on the United States to provide direct military assistance to his semi­autonomous region, which he complained has been left to fight the extremists unaided.

For the Kurds, the Islamic State’s blitz across northern Iraq in recent weeks has in some respects been a boon. Iraqi military forces­ rapidly withdrew from the north, enabling the Kurds to seize areas where they had disputed control with Baghdad, including the oil-rich city of Kirkuk.

But the expanded borders mean more land to protect from their new, unwelcome neighbors. That added burden comes as the Kurdish region’s relations with Baghdad are sharply deteriorating, with the central government withholding funds, banning cargo flights and making it difficult for the Kurds to replenish their arms.