Turkish soldiers head to Syria from the border city of Karkamis in Turkey’s southern region of Gaziantep on Saturday. (Bulent Kilic/AFP via Getty Images)

Clashes between Syrian rebels and Kurdish-aligned forces, both backed by the United States, intensified Sunday in northern Syria, as the rebels seized villages from the Kurds and Turkish warplanes pounded Kurdish positions, killing dozens.

The fresh fighting suggested that Turkey and its Syrian proxies are increasingly focused on stopping Kurdish forces from gaining more territory in northern Syria, particularly along Turkey’s border, potentially signaling a widening of the conflict.

And it threatens to divert resources and attention away from the campaign against the Islamic State. At the same time, Turkey, a NATO ally, risks fueling friction with Washington, which views the Kurds and their allies as the most effective U.S. partners against the Islamic State.

The outbreak of fighting highlights the virtually impossible choices the United States faces in supporting forces that are mutually hostile — from among the Turks, the divided non-Islamist Syrian opposition, and the Kurds.

“We want to make clear that we find these clashes unacceptable and they are a source of deep concern,” Pentagon spokesman Peter Cook said Sunday. “This is an already crowded battle space. Accordingly, we are calling on all armed actors to stand down immediately and take appropriate measures to de-conflict.”

Turkish warplanes and artillery struck targets held by the Kurdish YPG militia in north Syria on Aug. 28, security sources said, as Turkey pressed on with the cross-border campaign it launched last week with its Syrian rebel allies. (Reuters)

Uncoordinated operations, Cook added, will give Islamic State forces, which remain “a lethal and common threat,” the opportunity to find sanctuary.

Sunday’s clashes came a day after a rocket attack on two Turkish tanks killed a Turkish soldier and injured three others in northern Syria. Turkey, which is wrestling with Kurdish insurgents within its borders, blamed the attack on Kurdish forces. The casualties were Turkey’s first since it dispatched tanks and special forces units, backed by U.S. and Turkish fighter jets, into Syria on Wednesday to oust Islamic State militants from the border town of Jarabulus.

The militants fled the town without putting up a fight. Since then, Syrian rebels have been pushing westward, chasing the Islamic State, as well as southward into areas controlled by forces aligned with the U.S.-backed Syrian Democratic Forces. The alliance is largely dominated by the Syrian Kurdish People’s Protection Units, or the YPG, but also includes some Arabs.

[Turkey’s Syria offensive is as much about the Kurds as ISIS]

On Sunday, pro-Turkey Syrian rebels of the U.S.-backed Free Syrian Army said they had wrested 10 villages from Kurdish control, while seizing four villages from the Islamic State, also known as ISIS or ISIL. A video posted on social media showed Syrian rebels beating captured fighters allied with the Kurds.

Turkey’s state-run Anadolu news agency said Sunday that Turkish airstrikes killed 25 Kurdish “terrorists” and destroyed five buildings from where the fighters were firing at advancing rebels.

The Syrian Observatory for Human Rights, a Britain-based monitoring group, said Turkish airstrikes and artillery shelling killed at least 20 civilians and wounded dozens during a fierce overnight battle for a village. It was unclear whether the Turks and the monitoring group were referring to the same incident.

[Biden warns Kurds not to seek separate enclave on Turkish-Syrian border]

The YPG’s senior command said in a statement that it was not engaging Turkish forces “despite the losses we suffer.” It added that “to stabilize the north of the country, the goal remains fighting Daesh and not Turkish forces,” using an Arabic acronym for the Islamic State.

Both Turkey and Syrian rebels say the YPG has been targeting their forces. They say the Kurds broke a pledge to move their forces east of the Euphrates River, which senior U.S. officials also demanded, and are pressing for more terrain. The YPG insists that it has pulled back its forces. What is clear, though, is that its allies have not.

Shervan Derwish, a spokesman for a Kurdish-aligned military council in Manbij, 25 miles south of Jarabulus, said Sunday that the “battles are still ongoing.” At least 20 to 25 Turkish airstrikes have hit areas between Manbij and Jarabulus since Saturday, he said.

“Turkey didn’t come to fight Daesh, they came to fight us,” said Derwish, who is an ethnic Kurd and served last year as the spokesman for Kurdish forces in the Syrian town of Kobane.

[Turkey’s Syria offensive may have been delayed a year by officers who plotted coup]

Turkey’s concerns about Kurdish expansion grew after the Kurdish forces drove the Islamic State from Manbij this month and then began pushing north toward Jarabulus. Turkey’s incursion last week, code-named Euphrates Shield, preempted the Kurds from seizing the town.

The Turkish government is worried that Kurdish aspirations for a corridor linking two Kurdish enclaves in northwestern Syria could lead to an independent Kurdish state along its borders. That, Turkey fears, could embolden Kurdish militants on its own soil who have been a waging a three-decade-long armed struggle for cultural and political rights and self-determination.

At a rally Sunday in the southeastern city of Gaziantep, about 30 miles from the Syrian border, Turkish President Recep Tayyip Erdogan declared that he was committed to fighting the Islamic State, but he also vowed to wipe out the main Syrian Kurdish Democratic Union Party, whose military wing is the YPG.

“We will continue until we uproot this terror organization,” Erdogan told the thousands gathered at the rally.

Hamil Harris in Washington and Zakaria Zakaria contributed to this report.

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Today’s coverage from Post correspondents around the world