On the night of Jan. 30, the Islamic State used the cover of fog to attack a Kurdish battalion near the town of Mahmour. Seven Kurds were killed immediately. Their colleagues said that if they had had night-vision goggles — or better yet, thermal-imaging scopes to also detect vehicles — all would most likely be alive. When we gave them a gift of our small, store-bought binoculars with which we had been watching Islamic State movements less than one mile away, they expressed deep gratitude. As we left, a mine went off as they moved earth to make a defensive wall, for there is no demining equipment.

To be sure, coalition airstrikes have prevented Islamic State forces from deploying heavy artillery to break Kurdish lines, although Gen. Sirwan Barzani, who commands the main front between Erbil and Mosul, told us that a Pentagon lawyer must approve every strike (a policy intended to minimize chances of civilian casualties from drone attacks). Sometimes, that approval comes too late.

With its big guns vulnerable to air attack, the Islamic State adapts its tactics, piercing Kurdish lines with suicide attacks in primitively armored vehicles. One Kurdish commando near the Mosul Dam showed us, on his smartphone, a video of the approach of a steel-hardened vehicle. No amount of rifle fire or rocket-propelled grenades could stop the attack, which killed 23 and wounded 40.

Yet the United States insists that the Kurds obtain permission, grudging and often denied, from the central government in Baghdad for essential equipment to counter these and better weapons that the Islamic State seized from the Syrian and Iraqi Armies.

Meanwhile, the Islamic State won’t quit. Their wounded fighters often booby trap their bodies rather than be captured, and face down fire to recover dead comrades’ bodies. The leaders they call emirs, who are chosen because of their religious devotion and fearless effectiveness, and their foreign fighters, are especially fierce. The Westerners often die in suicide attacks; seasoned fighters from North Africa and the Middle East, and particularly from former parts of the Soviet Union (like Chechnya, Uzbekistan, Dagestan), are prominent as operational leaders and snipers. Foreign fighters return to their countries only if they escape or are sent home, because the punishment for defection is death.

Local Syrians and Iraqis conscripted to fight for the Islamic State, in contrast, are not totally committed. In one conversation picked up by a Kurdish walkie-talkie, a fighter with a local accent asked for help: “My brother has been killed. I am surrounded. Help me take his body away.” The reply: “Perfect, you will join him soon in Paradise.” The fighter retorted: “Come for me. This Paradise, I don’t want.”

The Islamic State will say to a local sheikh: “Give us 20 young men or we loot your village.” To a father with three sons, they will say: “Give us one or we take your daughter as a bride for our men.” One girl of 15 told how she was “married” and “divorced” 15 times in a single night to a troop of Islamic State fighters (under some readings of Shariah law, “divorce” is as easy as repeating “I divorce you” three times, which makes it easy to cast rape as marriage). In the face of such brutality, wavering supporters of the Islamic State could well rally to an Arab Sunni force allied with the Kurds. That is a prospect the United States, which fears leaving the fight mainly to Iran and its allies, should welcome.