A video posted on social media showed Syrian rebels beating captured fighters allied with the Kurds.
Turkey's state-run Anadolu news agency said that Turkish airstrikes killed 25 Kurdish "terrorists" and destroyed five buildings 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.
The increased tensions between the CIA and Pentagon-backed Syrian rebels and Kurdish forces threaten to take resources and attention away from the campaign against Isis. Targeting the Kurdish forces also could fuel friction with Washington, which views the SDF and the YPG as its most effective partners against Isis.
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 stabilise the north of the country, the goal remains fighting Daesh and not Turkish forces", using an Arabic acronym for Isis.
Both Turkey and Syrian rebels say the YPG, as part of the SDF, has been targeting their forces. They say the Kurds broke a pledge to move their forces east of the Euphrates River, which senior American officials also demanded, and are pressing for more terrain. The YPG insists that it has pulled its forces back. What is clear, though, is that its SDF allies have not.
Shervan Derwish, a spokesman for a Kurdish-aligned military council in Manbij, said that the "battles are still ongoing".
At least 20 to 25 Turkish airstrikes have hit areas south of Jarabulus since Saturday, he said.
"Turkey didn't come to fight Isis, 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 concerns about Kurdish expansion grew after the SDF drove Isis from Manbij this month and then began pushing north toward Jarabulus. Turkey's incursion last week pre-empted the Kurds from seizing the town.
Ankara 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 PKK 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 yesterday in the southeastern city of Gaziantep, about 50km from the Syrian border, Turkish President Recep Tayyip Erdogan declared that he was committed to fighting Isis, 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 organisation," Erdogan told the thousands gathered at the rally.