Iraqi forces stand guard at the Mosul dam in northern Iraqi on Tuesday. (Ahmad al-Rubaye/AFP via Getty Images)

Iraqi forces backed by Iranian-allied militias began an assault Thursday to reclaim more Kurdish-held territory in Iraq, advancing toward a crossing in the country’s western border region that provides the only access for U.S. military operations in northern Syria.

A protracted fight over border crossings could severely disrupt U.S. military activity in neighboring Syria. It also could strain the ability of aid organizations to provide desperately needed supplies to the nearly 300,000 civilians who fled fighting in the Syrian city of Raqqa, which U.S.-backed forces reclaimed from the Islamic State militant group this month.

The fresh assault in northern Iraq came as Prime Minister Haider al-Abadi dismissed a Kurdish offer for a cease-fire amid tensions over the Kurds’ referendum last month in favor of independence.

The clashes signal that Baghdad is determined to follow through on its goal of exerting full control over all of Iraq’s borders, including those crossings in the north that had been operated by the semiautonomous Kurdish regional government and have served as an economic lifeline for it.

The combined federal police and militia forces set out at dawn from the town of Zumar, north of Mosul. The plan appeared to be to gain control of the Fishkhabour border crossing from Kurdish fighters, known as peshmerga, and then sweep toward the main border crossing with Turkey — which for decades has been the most important outlet for Kurdish commerce.

Col. Ryan Dillon, a spokesman for the U.S.-led international coalition against the Islamic State, said the fighting has hampered its efforts to defeat the group, citing the inability to move military equipment and supplies to allied forces in Iraq and Syria.

Dillon said that the majority of the flights carrying humanitarian supplies into Syria have not been disrupted but that the transport of heavy military equipment that cannot be flown in has. This stems from an inability to coordinate with Iraqi and peshmerga senior officers, who have been unavailable because of the ongoing fighting, Dillon said.

Speaking during a visit to Iran on Thursday, Abadi said he would not accept anything less than a full annulment of the Kurdish referendum. The vote set off the ongoing crisis, in which Iraqi forces have entered disputed ­areas for the first time since 2014. Some of the areas Iraqi troops are moving into have been controlled by Kurds since 2003.

Kurdish leaders offered Wednesday to “suspend” the referendum results in exchange for Iraqi forces’ stopping their advance.

In an interview Tuesday with The Washington Post, Abadi reiterated that asserting Iraqi government control over disputed areas included all borders.

“The borders should be run by the federal state,” he said. “This is the exclusive authority of the Iraqi and federal government to do this, and we intend to complete this task.”

The Kurdish region’s security council issued an urgent plea late Wednesday for intervention by the U.S.-led coalition to halt the “unprovoked” Iraqi government assault, disputing Baghdad’s claims that movements were ­being coordinated with Kurdish authorities.

“Baghdad should withdraw all forces from nearby areas and accept KRG’s offer for unconditional talks to settle political differences,” the council said in a statement, referring to the Kurdish regional government.

Ammar al-Jazairi, a spokesman for the Emergency Response Division (ERD), an elite SWAT unit of the Iraqi federal police, said the pro-Baghdad forces have not been given orders to fire at peshmerga fighters and are moving slowly toward the area of Fishkhabour to avoid clashes.

“We are waiting for them to retreat,” he said. “We are a major force, but we want to take it peacefully because they are still Iraqis. Therefore, we are waiting for them to retreat, but if they insist on fighting, then we will have no choice and will take it by force.”

Jazairi said peshmerga forces have attempted to stop the advance using mortars and rockets, killing at least seven members of the ERD and federal police as of late morning Thursday. Kurds,
in turn, claim that they have destroyed three tanks, five U.S.-made Humvees and an armored vehicle.

Jazairi said the pro-Baghdad force is composed of the ERD, federal police and allied Shiite militias, which operate under the command of the central government. The militias, known as popular mobilization units, worry both the Kurds and the United States, which views them as being closer to Tehran than to Baghdad.

In one of the contradictions of U.S. regional policy, the United States is fighting on the same side as the Iranian-backed militias in Iraq but on the opposite side in Syria, where it has clashed with some of the same groups.

If Fishkhabour falls to those militias, it would be a “grievous blow” to the United States’ Syria policy, said Nicholas Heras of the Center for a New American Security, a Washington think tank.

Heras said it could jeopardize U.S. operations against the Islamic State in Syria and plans to stabilize Raqqa.

“Fishkhabour is more than just a border gate: It is the essential doorway for long-term American influence in Syria. And Iran wants to slam that gate shut,” he said.

The battle for the Fishkhabour crossing also risks throwing an already stretched aid operation for displaced Syrians into disarray. More than 270,000 people have fled Raqqa and filled makeshift camps in northeastern Syria since June, when the U.S.-backed offensive to capture the city began. Aid officials said Thursday that those camps are heavily reliant on supplies coming through Fishkhabour.

The struggle between the central government and the Kurds has nearly overshadowed the remainder of Iraq’s fight against the Islamic State, which has been the primary fear of the United States. 

To little fanfare, Iraqi forces Thursday launched an offensive in the southwest to push Islamic State militants out of their last stronghold in the country. Iraqi forces began moving on Qaim, a town that borders Syria’s eastern Deir al-Zour province. The U.S. military suspects that the border area between Iraq and Syria is the hiding place of the Islamic State’s leader, Abu Bakr al-Baghdadi.

Sly reported from Beirut. Louisa Loveluck in Beirut contributed to this report.