Turkey has urged the US to stop backing the Kurdish YPG in Syria, as it steps up an offensive against the militia.
A spokesman for President Recep Tayyip Erdogan told the BBC that Kurdish fighters were using US-supplied weapons against Turkish troops trying to oust them from the Afrin region.
Turkey considers the militia a terrorist group, and an extension of the Kurdistan Workers Party (PKK) which wants Kurdish autonomy within Turkey.
The YPG denies any direct links.
The militia, which controls much of north-eastern Syria, has been a key US ally in the fight against Islamic State (IS) fighters there.
Ankara is now demanding an end to the alliance, arguing that the fight against IS is over.
"We cannot tolerate the PKK establishing some kind of a state structure along our border in Syria," warned presidential spokesman Ibrahim Kalin.
Thousands of civilians are reportedly trying to flee Afrin, and Syrian activists say more than 70 people have died since the Turkish push began on Saturday.
The UN Security Council discussed the growing offensive in a meeting on Monday, but did not condemn it.
US Secretary of State Rex Tillerson said his country was willing to work with Turkey to address its "legitimate" security concerns in northern Syria.
He said the US recognised Turkey's right to defend itself from terrorist elements, and had proposed measures to try to stabilise the situation.
France's UN Ambassador Francois Delattre said Afrin "was of course part of the conversation" at the closed-door talks in New York.
"The call for restraint, I believe, was widely shared during the discussion," he added.
Earlier Turkish President Erdogan had vowed to "sort out" Afrin.
"We will take no step back," he said in a live television broadcast. "We spoke about this with our Russian friends; we have an agreement."
What has happened on the ground?
On Monday, the Afrin villages of Shankal, Qorne, Bali and Adah Manli were reportedly captured, along with rural areas including Kita, Kordo and Bibno, according to Turkey's state-run Anadolu news agency.
The YPG has denied that any of the Afrin region is under Turkish control.
It also said that it had retaliated against Turkey's ground offensive with rocket fire on Turkish border areas. Two people died and 12 others were wounded when a rocket hit a camp housing Syrian rebel fighters near the border in Turkey's Hatay province, local reports said.
What's the background to the offensive?
Turkish ground troops, assisted by rebels from the Free Syrian Army (FSA), crossed into northern Syria on Sunday as part of "Operation Olive Branch" - an offensive to push out the YPG.
Turkish Prime Minister Binali Yildirim said the aim was to establish a 30km (19-mile) "safe zone" deep inside Syria and Mr Erdogan vowed to crush the YPG "very quickly".
Plans for the operation are believed to have accelerated when US officials said earlier this month that it would help the Syrian Democratic Forces (SDF) alliance, which is dominated by the YPG, build a new "border security force" to prevent the return of IS.
Some 25,000 pro-Turkey fighters have joined the offensive, rebel commander Maj Yasser Abdul Rahim told Reuters. It is not clear how many Turkish soldiers are on the ground.
What are the casualties?
The Syrian Observatory for Human Rights, a UK-based monitoring group, said on Monday that 54 combatants had been killed since Saturday, including 26 Kurdish fighters and 19 pro-Turkish rebels. Twenty-four civilians had also died, it added - 22 in Turkish air and artillery strikes, and two as a result of Kurdish fire.
The Kurdish ANHA news agency reported earlier that at least 17 civilians have been killed in Turkish attacks.
Turkey said one of its soldiers had been killed.
Turkish Foreign Minister Mevlut Cavusoglu has denied killing civilians, and accused the YPG of emanating "nonsense propaganda and baseless lies".
How have the key players reacted?
The Syrian government, its ally Iran and Egypt have all condemned Turkey's offensive.
Western powers, including the UK and France, have urged restraint in order to avoid civilian casualties.
In earlier reaction:
Syrian President Bashar al-Assad condemned the incursion, saying Turkey was supporting terrorism.
Russia - a key ally of President Assad - also said it was concerned by the news, and withdrew some of its troops based in the area.
Iran, another Syria ally, called for a quick end to the operation "to prevent a deepening of the crisis" in Syria
The European Union's diplomatic chief Federica Mogherini said she was "extremely worried" by the offensive.
