"All the Turkish military's ground attacks against Afrin have been repelled so far and they have been forced to retreat", says YPG official Nouri Mahmoudi, according to Reuters.
President Bashar al-Assad on Sunday condemned Turkey's offensive against Kurdish militia in the northern Syrian region of Afrin, saying it was part of Ankara's support for extremist groups.
Also on Saturday, Syrian government forces and supporting militias attacked a rebel-held air base deep inside what was once opposition territory in northwest Syria.
Ankara fears the establishment of a Kurdish corridor along its border.
The attacks follow weeks of warnings against the YPG in Syria from President Tayyip Erdogan and his ministers. He said it would "clear our land up to the Iraqi border" of what he called "terror filth that is trying to besiege our country".
Turkish artillery and warplanes pounded YPG sites around Afrin and 153 targets, including YPG refuges and weapons stores have been hit, according to the army. The U.S. backs the YPG, even though Turkey, an ally, considers it a terrorist group.
Then when the Americans recently announced plans to create a 30,000-strong border security force mainly made up of Kurdish fighters, Turkey ramped up the rhetoric and has now matched it with on-the-ground action. The announcement has provoked outrage in Turkey.
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Syria's government had vowed to shoot down any Turkish fighter jets over Afrin, calling it an "aggressive act".
"France asks for an urgent meeting of the Security Council", French Foreign Minister Jean-Yves Le Drian tweeted, adding he spoke to his Turkish counterpart Mevlut Cavusoglu by phone on Sunday.
"Hopefully, we will complete this operation in a very short time", Erdogan said in a speech in the western city of Bursa.
Last week the US government urged Turkey not to attack the YPG. "We call on the opposing parties to show restraint".
The foreign ministry of Russian Federation, whose opinion is seen as crucial in determining how far the operation goes, said it had received the news of the campaign with concern and urged restraint.
He added, "The Islamic Republic of Iran attaches importance to protecting Syria's territorial integrity and respecting this country's national sovereignty".
Shahoz Hasan, head of Syria's main Kurdish political party, told VOA Saturday that the Turkish operation could get in the way of a political solution in Syria. The Afrin district houses no less than 800,000 civilians, including displaced people from earlier years of the Syrian war.