Turkey considers the YPG a "terrorist" group and said today it would continue to target the group if it failed to retreat east of the Euphrates River.
The clashes bolster Western concerns that Turkey's military incursion into Syria is intended, in part, to target US -supported Kurdish forces known as the Kurdish People's Protection Units, the YPG militia.
U.S. Defense Secretary Ash Carter says the United States is trying to "de-conflict" U.S. - backed actions against the Islamic State by Turkey and by the American-backed Syrian Kurdish rebels known as the Syrian Defense Forces.
But Turkey's incursion into Syria also had the dual goal of making sure the YPG didn't capture Jarabulus first.
PYD leaders say they have withdrawn, but their units advise the Syrian Democratic Forces, and it is not clear if any remain west of the Euphrates.
Turkey's president has said his country will press ahead with its military operation in Syria, despite the United States expressing "deep concern". Accordingly, we call on all armed actors to stand down and take appropriate measures to deconflict & open channels of communication.
Ankara is anxious that Kurdish forces are trying to extend their area of control along Turkey's southern border. Turkey denied any civilians had been hit.
Now the rebels have retaken many villages south of Jarablus, the group backed by Turkey is advancing towards the city of Manbij a YPG stronghold in northern Syria.
ISIL is an acronym for the Islamic State (IS) group.
There were also reports of clashes on the ground between Turkish forces, Turkish-backed rebels groups and Kurdish forces.
A U.S. defense official said the US-backed Kurdish forces had pulled back to east of the Euphrates river over the past day or so, as demanded by Ankara.
Cook said the US doesn't support reported Turkish airstrikes and artillery shelling of USA -backed Syrian Kurdish fighters - or Kurdish attacks on Turkish troops - in areas where IS fighters no longer operate.
Smoke rises from damaged police headquarters after a suicide truck bombing killed eleven Turkish police officers and injured 78 people on August 26, 2016 in Cizre, southeastern Turkey.
Turkey's use of force against Kurdish forces in Syria has not gone down well with the USA, with Washington openly supporting the Kurdish YPG fighters, who have proved to be a vital ground force in the battle against Islamic State (IS, formerly ISIS/ISIL).
"The United States was not involved in these activities, they were not coordinated with U.S. forces, and we do not support them", he said.
"We declare the withdrawal of our forces to south, to the Sajour River, to preserve the lives of civilians and so that they (Turks and their allies) don't have any justification to continue shelling civilians", the council's statement read.
On Monday, Turkish-backed forces advanced on Manbij, a city about 30km (20 miles) south of Turkey's border captured this month by the SDF with U.S. help.