But the campaign risks further increasing tensions with Turkey's North Atlantic Treaty Organisation ally Washington, which has supported the YPG in the fight against Islamic State (IS) jihadists and warned Ankara about distracting the focus from that fight.
"All the Turkish military's ground attacks against Afrin have been repelled so far and they have been forced to retreat", Nouri Mahmoudi, a YPG official, said.
"We call on all parties to remain focused on the central goal of defeating" IS, State Department spokeswoman Heather Nauert said in a statement.
Turkish troops on Sunday crossed over the Syrian border into the Afrin region on the second day of Ankara's operation against the Peoples' Protection Units (YPG) Kurdish militia, Prime Minister Binali Yildirim said.
Turkey's military has been shelling the Afrin region since Thursday, a move which it said was in response to fire coming from the area.
State-run Anadolu news agency said that Turkish troops, accompanied by pro-Turkey rebels from the Free Syrian Army, had already advanced some 5km (3 miles) into Syrian territory. It has been deploying troops and tanks to the border in past weeks.
The Britain-based Syrian Observatory for Human Rights says the Kurdish militia and Turkish forces have clashed on the northern and western edges of Afrin, but that Turkish troops have failed to advance.
The YPG has also accused Turkey of striking civilian districts and a camp for the displaced in Afrin.
Turkey's foreign minister, Mevlüt Çavuşoğlu, discussed the operation with Tillerson by phone after the U.S. diplomat requested a conversation, Turkish officials said. "There will be a heavy price to pay by those who respond to this call", he added.
US Oil Production to Hit New Record in 2018
This, combined with healthy oil demand, has pushed up crude by nearly 15 percent since early December. Distillates - used to make diesel and heating oils - fell by 3.9 million barrels.
Major Yasser Abdul Rahim said the rebels did not seek to enter the mainly Kurdish city of Afrin but encircle it and expel the YPG, which controls it.
Meanwhile, tens of thousands of Syrians in the northwest are fleeing the renewed violence amid a new government offensive in neighbouring Idlib, converging on the Turkish border and igniting fears of a new wave of migration.
Earlier in the day, Turkish media reported four rockets fired from Syria had hit the Turkish southern border town of Kilis, wounding one person and causing damage to houses there. Turkey regards the YPG as the Syrian arm of the Kurdistan Workers' Party, or PKK, despite the group's help in battling ISIS.
He was referring to the Kurdish Democratic Union Party and its armed wing, the YPG, fighting in Syria. "It calls on the Turkish authorities to act with restraint in a hard context where the humanitarian situation is deteriorating in several regions of Syria as a result of the military operations carried out there by the regime of Damascus and its allies", the Ministry said in a statement.
Yildirim was quoted as saying that the Turkish forces aimed to create a security zone some 30 kilometres (18 miles) deep inside Syria.
Russia's foreign ministry has said it is concerned by news of the offensive, and is urging restraint.
The Syrian government urged the worldwide community to condemn the attacks and take steps to end this "aggression", the state-run Syrian Arab News Agency reported, citing the country's foreign ministry.
But the Syrian foreign ministry strongly denied this and President Bashar al-Assad slammed the offensive as "support for terrorism".