MUFTI, Iraq — The pickup trucks on their way to this village in northern Iraq on Sunday kicked up the dry earth on the dirt track, clouding the air and limiting the visibility for the drivers approaching the hamlet just wrested from the so-called Islamic State with the help of elite American soldiers operating now in both Iraq and Syria.

Engines roared as the cars accelerated to avoid getting stuck in the loose earth, drowning out the drone of coalition warplanes circling above in the gradually building offensive to retake Mosul, Iraq’s second-largest city, conquered by ISIS two years ago.

Then, without warning, a sharp explosion tore through the air, and a column of smoke billowed upwards. The fighters milling around a school building at the edge of the village barely took note—this was one of nine suicide attacks the Kurds had to fend off during the day’s fighting. Previous engagements had set the arid fields on fire, and pillars of smoke reached for the sky all around Mufti.

The biggest threat to the Kurdish fighters in the past have been ISIS vehicles laden with explosives and protected by sheets of metal welded onto their fronts and sides. This suicide bomber was in one of them, but before he could get close, the Peshmerga hit him with a Milan anti-tank missile, donated to the Kurds by Germany.

The Milans featured prominently in this little battle, while other suicide attacks were stopped by airstrikes directed by the American soldiers on the ground.

A once-dreaded weapon capable of tearing huge holes in enemy defenses, the so-called vehicle borne improvised explosive device (VBIED) is at the mercy of the Western war effort supporting the Kurds at Mufti, a village in the Khazir section of the front on the approach to Mosul.

A steady stream of vehicles laden with heavily armed Kurdish Peshmerga fighters have poured into the village since it was taken from its ISIS defenders on Sunday morning, and it has since become a staging area for the unfolding offensive that continued into Monday.

The operation is the largest by the Kurds in Iraq since they took Sinjar from the Islamic State last November. Intent on driving ISIS out of nine villages facing them at the Khazir front, the Kurdistan Regional Government (KRG) threw 4,700 men into the offensive, according to Arif Tayfor, the sector commander at Khazir.

By Monday afternoon, seven of those nine villages had been taken.

The Kurds, without question, benefitted from some hands-on U.S. support. A few miles from Mufti, on the road leading directly to Mosul, I came across a U.S. special operations commando shoveling empty machine-gun cartridge cases out of the turret of an armored car.

These camera-shy elite soldiers usually refrain from engaging the enemy directly, instead gathering intelligence and directing air strikes. But at Khazir, U.S. ammunition clearly was expended.

It is not the first time American special operations forces have tangled with ISIS on the Kurdish front lines in Iraq. Early in May, U.S. Navy Seal Charlie Keating was killed when a group of Seals helped contain an ISIS attack on Telskuf, an abandoned Christian town near Mosul.

The Khazir operation’s immediate aim is to relieve the pressure on the nearby frontline town of Gwer and push ISIS further away from Erbil, the capital of the autonomous Kurdish region that is barely an hour’s car ride away.

The long-term goal is to carve out a greater Kurdistan from the crumbling caliphate and a disintegrating Iraq. The villages at Khazir are part of the disputed territories, areas claimed by both the KRG and the central government in Baghdad.

The Kurds feel that these territories, which form a broad swathe of land straddling the Kurdish region from the Iranian to the Syrian border, are rightfully theirs, and were taken from them through resettlement and the redrawing of administrative borders under the Iraqi tyrant Saddam Hussein.

Until this offensive, the villages at Khazir had formed part of the protective belt shielding ISIS from Peshmerga encroachment on Mosul.

The Iraqi government and the international coalition supporting the fight against the terror group are desperate to kick it out of Mosul and so put an end to ISIS in Iraq.

The Kurds share this goal, but are equally keen to take control of the areas to the city’s north.

The offensive at Khazir is in line with their ambitions. By pushing ISIS out of the area, the Peshmerga are creating facts on the ground.

“These are Kurdish villages that we control, not Iraqi villages. According to Article 140 of the Iraqi constitution, these are disputed areas,” sector commander Tayfor tells The Daily Beast at his headquarters near the front.

He is referring to a provision in the Iraqi constitution for a referendum in the disputed areas, something that Iraq’s malfunctioning political system has made no progress towards carrying out.

“When we control of all of Khazir district and all of the area around Gwer... we get rid of this debate about Article 140,” he adds.

Tayfor insists that the inhabitants of the disputed areas will be given a choice whether they want to become part of the Kurdish region or not, in the full knowledged that becoming part of Kurdistan could mean an eventual departure from Iraq.

KRG President Masoud Barzani in February announced that he intends to hold a non-binding referendum on whether the Kurdish region should secede from the rest of the country. He later added that the plebiscite should be held before the U.S. elections in November.

According to Tayfor, the inhabitants of the disputed territories would be included in that referendum, giving them a say in their future.

Kurdish officials have voiced the idea before.

Najmaldin Karim, the governor of Kirkuk, earlier this year called for the city to be included in a Kurdistan-wide referendum to determine if it becomes part of the KRG. Oil-rich Kirkuk—the biggest prize in the disputed territories—was occupied by Kurdish forces to prevent it from falling to ISIS in 2014.

The villages being fought over at Khazir were inhabited by Kakai and Shabak religious minorities, who identify as Kurds. Further north, towns such as Bashiqa were home to a Christian population before it fled ISIS, while still further north lies Sinjar, where the Yazidi religious minority fled the onrushing jihadists in August 2014.

After Bashir, the Kurds will turn their attention to the Christian areas that border Mosul.

While the Kakai and the Shabaks will likely seek to become part of Kurdistan, the preferences of the Christians and the Yazidis are not yet clear. But should they vote to be incorporated into the Kurdish region, there is little the faltering central government in Baghdad can do to prevent it once the Peshmerga are in control.

Baghdad has also been forced into some highly unfavorable horse trading over Kurdish support for the battle to retake Mosul from ISIS. The Iraqis need all the help they can get to expel the terror group from the city, but the Kurds are not going to sell themselves cheaply.

“If we don’t reach an agreement, we won’t take part in the liberation,” says Tayfor decisively. The ageing commander is part of the old guard of Peshmerga that spent years fighting Saddam’s soldiers in the mountains of Kurdistan, and who rose up against the dictator during the First Gulf War in 1991, only to be crushed when the Americans failed to come to their aid.

Over the past two years, the Kurds have been fighting aggressors from the south once again. The sectarian politics pursued by former Prime Minister Nouri al Maliki had succeeded in growing the ranks of ISIS, which duly attacked the Kurdish region. Some of the terror group’s military commanders are revanchist former officers in Saddam’s army.

Faced with a dysfunctional Iraq, many Kurds want out. As Kurdistan begins to eye an exit from Iraq, it intends to take with it as many of the disputed areas as possible.