SINJAR, Iraqi Kurdistan—The crash of the incoming mortar came from way behind the Kurdish lines; the shells were landing in the rear. Yet the excited and somewhat fearful commotion among the peshmerga fighters was instant. Men were looking through the peek holes at the Islamic State (IS) lines a few hundred yards away, trying to locate the mortar. Kurdish officers were on the phone, calling for a coalition airstrike.

The warplane soon came roaring in, but by then the mortar had disappeared among the houses. A second coalition jet targeted an IS fighting unit—the next, a tank. The Kurds rely on these airstrikes a great deal. In the two days I spent on the frontline with them earlier this month, they called in about a dozen, trying to counter repeated assaults by the jihadis from the Kurdish positions on the hills overlooking the city. “We are grateful to the coalition,” Col. Delgash Zebari, deputy commander of the 12th Peshmerga Brigade, told me. “If it weren’t for their help, maybe we couldn’t hold these areas.” ...