Washington considers the YPG its closest ally in Syria, viewing it as the most effective ground force in the fight against the Islamic State of Iraq and the Levant (ISIL, also known as ISIS) group.
Heather Nauert, spokeswoman for the US Department of State, said on Sunday that Secretary of State Rex Tillerson spoke with his Turkish and Russian counterparts on the phone and called against an escalation in fighting in northern Syria.
"We urge Turkey to exercise restraint and ensure that its military operations remain limited in scope and duration and scrupulous to avoid civilian casualties," said Nauert.
The statement on Sunday came just hours after Turkey said its ground troops had crossed into northern Syria.
Ankara considers Syria's Kurdish Democratic Union Party (PYD) and its armed wing, the YPG, "terrorist groups" with ties to the banned Kurdistan Workers' Party (PKK), which has waged a decades-long fight inside Turkey.
It fears the establishment of a Kurdish corridor along its border.
Speaking in Istanbul earlier on Sunday, Binali Yildirim, Turkey's prime minister, said Turkish forces had crossed into the YPG-controlled region in Syria at 08:05 GMT from the Turkish village of Gulbaba.
He said Turkey intended to establish a 30km "safe zone" in Afrin.
"Throughout the day we've heard the sound of jets in the sky, intense artillery and machine-gun fire," Al Jazeera's Stefanie Dekker, reporting from the Turkey-Syria border, said.
The Turkish army has said it is targeting only "terrorist" positions and "utmost importance" was being given to not harm civilians.
According to estimates, there are between 8,000 to 10,000 Kurdish YPG fighters in the Afrin area.
"Turkey says it will continue its operation until it has pushed the YPG away from its borders," our correspondent said.
No one knows how long this will take or what the implications may be.
"The YPG are extremely well trained. They know the terrain in Afrin but Turkey has superiority in the skies, and that is a huge advantage."
Reacting to Turkey's ground advance into Syria, France called for an emergency meeting of the UN Security Council.
"Ghouta, Idlib, Afrin - France asks for an urgent meeting of the Security Council," Jean-Yves Le Drian, the French foreign minister, said on Twitter on Sunday.
Le Drian said he had talked to his Turkish counterpart during the day and called for a complete ceasefire in Syria.
France's request for the emergency meeting comes one day after the Turkish operation against the YPG began.
On Saturday, Turkish jets carried out air raids against targets in Afrin.
Russia, which controls the airspace over Afrin, withdrew hundreds of its soldiers deployed near the city before Turkey's operation began.
Several diplomats and mission chiefs of the permanent members of Security Council - the US, Russia, the UK, France and China - were brought up to speed on the developments in Afrin.
For his part, Turkish President Recep Tayyip Erdogan said on Sunday he hoped the military operation would be completed "in a very short time".
But Erdogan also warned pro-Kurdish opposition supporters in Turkey not to hold protests.
"Know that if you go out on the streets, authorities are on your necks," he told thousands of supporters in Bursa.
"This is a national struggle, and we will crush anyone who opposes our national struggle."
Earlier, Anadolu news agency reported that Turkish-backed Free Syrian Army (FSA) fighters had advanced towards Afrin in the early hours of Sunday, supported by Turkish troops.
The YPG confirmed the advance, saying two villages in Afrin's Bilbil district near the Turkish border had come under attack.
About 25,000 Free Syrian Army (FSA) rebels were joining the Turkish military operation in northern Syria with the goal of recapturing Arab towns and villages held by the YPG, a Syrian rebel commander told Reuters news agency on Sunday.
As early as Friday, thousands of FSA fighters had already been mobilised in Turkey's Hatay province and Syrian locations east of Afrin.
Major Yasser Abdul Rahim, who is also the commander of Failaq al-Sham, a main FSA rebel group in the operations room of the campaign, said the rebels did not seek to enter Afrin but encircle it and expel the YPG.
"We have no interest in entering the city only the military targets inside the city and the villages around it. We aim to encircle the city and ensure the militias are evicted. We won't fight in the city as we have no problem with civilians," he said.
A main goal of the military operation was to recapture Tel Rifaat, a town southeast of Afrin, and a string of Arab villages the YPG captured from rebels in February 2016, driving out tens of thousands of inhabitants, Abdul Rahim said.
|Smoke rises from Syria's Afrin region following the Turkish operation on Saturday [Reuters]|
"The task of the Free Syrian Army is first to regain sixteen Arab towns and villages occupied by the foreign militias [YPG] with the help of the Russian air force," Abdul Rahim told Reuters in a phone interview from inside Syria.
The fighting Abdul Rahim was referring to forced at least 150,000 residents of these villages to flee to Azaz.
They are sheltering in camps at the Turkish border and rebels say they have not been allowed to go back to their homes.
On Saturday, Erdogan said the operation in Afrin would be followed by a push in the northern town of Manbij, which the US-backed Kurdish forces captured from ISIL in 2016.
Source: Al Jazeera News