In a brief statement, the Turkish military said that the new siege on Afrin had begun on Monday, adding that it had taken control of "critical areas" of the town without giving further details.
Birusk Hasakeh, a YPG spokesman inside Afrin, denied the city had been totally besieged but said the last route leading out of it was being shelled heavily.
The protesters were expressing anger at a perceived lack of action from the US after Turkey announced that its troops and Turkish-backed Syrian opposition fighters had begun a siege of the Syrian Kurdish-held northern town of Afrin.
Turkey has explained that it is battling Kurdish "terrorists" to justify the offensive in Afrin, which has been largely untouched by Syria's deadly conflict until recently. He noted that two militiamen from the pro-government forces had been killed in the first day after their arrival in Afrin, while three had been wounded.
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Turkey launched operation "Olive Branch" in January to clear the YPG from the Afrin enclave.
Pro-Turkish forces have seized several border areas and have nearly fully encircled the city of Afrin.
The military also said only terror targets are being destroyed and that "utmost care" is being taken to avoid harming civilians.
Pro-Kurdish demonstrators have staged protests in Paris against Turkey's military offensive in northern Syria and clashed with police outside the United States embassy, AP reported.
Thousands of people had started to flee Afrin on Monday as the Turkish troops got closer to the town, heading toward nearby government-controlled areas.
Turkey considers the Kurdish Democratic Union Party (PYD) in Syria and its armed wing YPG to be "terrorist groups" with ties to the banned Kurdistan Workers' Party (PKK).