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 stabilize 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.”