LONDON — The battle for Sinjar has begun.

Roughly 7,500 Kurdish forces, backed by U.S. airstrikes, began their long-anticipated offensive on the city early Thursday.

A barrage of strikes, followed by Kurdish militants storming into the city and vowing to the “fight to the death,” marked the start of the campaign as part of a multi-faction offensive. The ground contingent features troops from the Kurdish peshmerga paramilitary units, 1,500 Yazidis, an ethnic minority against whom ISIS is accused of committing genocide a year and a half ago, and militias loyal to both the Kurdish Workers’ Party (PKK) and its Syrian affiliate, which fights under the umbrella of the Self-Protection Units, or YPG.

The offensive to retake Sinjar is the latest military effort to weaken ISIS, this time by cutting off Highway 47, the main supply route that links the terror group’s caliphate headquarters in the northern Iraqi city of Mosul with its burgeoning state in eastern Syria. The Kurdish fighters may have taken as many as 35 miles of the highway away from ISIS and seized six villages on either side of Sinjar, although different factions are claiming credit for the operations. The peshmerga, beholden to the Kurdistan Regional Government in Erbil, is ideologically at odds with the PKK and YPG militias.

Nevertheless, the U.S. considers the Kurds the best local ground force against ISIS, are receiving support from the coalition in the air. For the past three days, the Pentagon has conducted 18 strikes. And in the hours leading up to the ground offensive, the U.S.-led coalition conducted 20 strikes around Sinjar. By Thursday morning, Kurdish troops moved in.

But the strikes also signaled to ISIS to get ready for a battle, and it appears the group has moved in reinforcements and set IED traps all around the city in preparation for the fight, according to witnesses on the ground.

Brigadier Salar Taymour from the al-Jarabani Division in Sinjar told The Daily Beast Thursday afternoon: “Sinjar is completely surrounded by our peshmerga fighters and we managed to fight on six different fronts at the same time. We haven’t been faced with an army in each village—we’ve faced with snipers and lots of suicide bombers.” ISIS, he added, is “hiding behind” car bombs.

The coalition conducted another 10 strikes after the offensive began, according to Department of Defense officials, which killed 66 and destroyed two car bombs. U.S. military officials said they believe it will take the Kurdish and Yazidi forces between two and four days of fighting to retake Sinjar, followed by another week of clearing the mountainous city.

“We have more than half a million Yazidi refugees who we want to have their homes back.”

What comes after remains unclear.

When Sinjar fell 18 months ago, the Yazidi minority religious group that lived there endured violent retaliatory attacks. Thousands of Yazidi men were killed; the women were captured and used as sex slaves. Thousands of others fled. The persecution of Yazidis prompted the U.S to re-enter Iraq for the first time since it withdrew troops in 2011, in a successful military campaign to rescue Yazidis trapped on the city’s mountain.

Many Yazidis hold the KDP responsible for failing to protect them from the ISIS onslaught and credit the PKK, which the U.S. considers a terror group, for helping them escape. Such a fraught recent history has many wondering how much the Yazidis will allow the KDP to reclaim in Sinjar. Will any territory go to the PKK instead?

U.S. military officials want to stretch ISIS in Iraq, weakening the group’s grip on much of western Iraq and eastern Syria. The plan is to force ISIS to defend their hold on Sinjar while simultaneously fending off the ongoing Iraqi security force push into the city of Ramadi, south in Iraq’s restive Anbar province, and the Kurdish led offensives in eastern Syria.

But ISIS has so far proven more agile than U.S. and Kurdish military planning and could find alternative supply routes between Mosul and Syria.

This story has been updated. With additional reporting by Mais al-Bayda.