Erbil, Iraqi Kurdistan — Fifty yards from my hotel, in the largely Christian neighborhood of Ainkawa, a car bomb went off last Friday. You can still smell the acrid smoke where the Nili cafe was shattered, killing two young men who had stepped out for a smoke not far from the U.S. consulate (which the terrorists didn’t reach).

But the Barista Coffee shop and Alreef Snack, just doors away, are open and patrons sit at outdoor tables. In fact, Erbil is one of the safest places in Iraq, despite far more security barriers than were visible before the Islamic State threatened the city last summer.

“Our security is very good considering all the threats,” says Nechirvan Barzani, the prime minister of the Iraq’s autonomous Kurdish region. That’s an understatement, given that Kurdistan has around 600 miles of the front line with Islamic State, and is less than 50 miles from Mosul, which the jihadis made the capital of their caliphate.

Kurdish peshmerga forces have made serious headway against Islamic State, taking back 90 percent of the Kurdish territory captured last year, as well as key areas outside their region. “We are holding strategic areas such as the Mosul dam because the Iraqi army is not capable,” said the urbane Kurdish leader, in an interview at his ornate headquarters in Erbil. “ISIS is on the defensive. U.S. air power has provided considerable help, for which we are grateful.”

So it’s hard to understand why the Kurds’ determined fight (and the Iraqi army’s continued weakness) hasn’t convinced Washington to give more direct support to the peshmerga. When it comes to breaking the Islamic State’s hold, the Kurds are the most reliable game in town.

One sign of the war’s toll is the ongoing flood of internally displaced Iraqis (along with Syrian refugees). Numbering 1.5 million, from all sects and religions, they have increased the Kurdish region’s population by nearly 30 percent and range from wealthy Sunni sheikhs from Mosul to Chaldean Christians to poor Sunnis from Tikrit. What really galls Barzani is that Baghdad barely contributes to the cost of helping the displaced.

Fly in a helicopter over northern Iraq, and you see sprawling, white-tented, refugee camps dotting lush green farmland. This puts an intolerable financial burden on the Kurdish government, when oil prices are low and budget aid from Baghdad is being withheld. Since the Islamic State’s attack, Kurdistan’s once-booming economy has stalled. Unfinished buildings dot the landscape.

I asked whether the Iraqi army would be ready to mount an offensive on Mosul any time soon. Barzani retorted, “No, I don’t think so. I don’t see any serious plan for it from Baghdad.”

In reality, retaking Mosul depends as much on Iraqi politics as military success. Islamic State-held areas are populated almost wholly by Sunnis who deeply distrust and fear the Shiite-led government. Unless they turn against Islamic State, it will be hard to expel the jihadis or prevent their return. “Our biggest question is what will happen the day after,” Barzani said.

Sunni fears are magnified by the fact that, apart from the Kurds, the strongest fighting forces in Iraq today are Iranian-backed Shiite militias. In the recent battle to take back Tikrit, these militias burned and looted Sunni homes.

Iraqi special forces units, backed by U.S. airstrikes, helped save the day in Tikrit, and barely kept the Sunni town of Ramadi from falling to the Islamic State last week. But Iraqi special forces are few and overstretched. Despite U.S. training efforts, most regular army units are far from ready for the fight.

Moreover, Iraq’s Shiite-led government is unwilling, with only minor exceptions, to arm those Sunni tribes that are willing to fight Islamic State. Barzani believes the Iraqi government must allow Sunnis to have their own autonomous region, which might make them more willing to deal with Baghdad. But that is a long-term project and doesn’t tell us who will expel the Islamic State from Mosul.

“The peshmerga can do it, if the issue is only military,” Barzani says. But, he adds, if peshmerga try to take the historic Arab core of Mosul, this will create strife between Kurds and nationalist Arabs. It isn’t something the Kurds are eager to do.

Meanwhile, the Kurdish fighters have to hold that 600-mile front. They remain the only large, reliable fighting force prepared to hold the Islamic State back, which makes it hard to understand why Washington fails to fully appreciate their critical role.

The United States insists that all military aid to the Kurds must be delivered via Baghdad, although it often doesn’t arrive or arrives too late or is insufficient. “What we do on the battlefront is not being understood or appreciated,” says Barzani. “We are on the front line fighting the worst terrorist organization the international community faces. We have lost 1,300 martyrs and 5,500 seriously wounded. We cannot sustain this.”

If the administration wants to see the Islamic State driven out of Mosul, the White House should be delivering aid directly to Erbil or pressing Baghdad to do so. Where’s the sense in penalizing the force with the best record of pushing the Islamic State back?