Turkey launched the cross-border offensive in January against the Syrian Kurdish YPG fighters who control Afrin.
U.S. officials have said they fear the Afrin offensive may divert attention from the anti-IS battle.
The UN chief said in a statement on Friday that he is "deeply concerned by the desperation shown by the people fleeing in a massive exodus from Eastern Ghouta and Afrin", Xinhua news agency reported. Tens of thousands of civilians have fled this battleground and the Turkish offensive in the Afrin theatre in the north, swelling the number of displaced over the past week to more than 270,000.
Civilians fleeing the city of Afrin in northern Syria walk at the mountainous road of al-Ahlam while heading towards the check point in az-Ziyarah, in the government-controlled part of the northern Aleppo province.
Since their January offensive began, Turkish forces have almost encircled Afrin in an effort to drive the Syrian Kurdish fighters from the town and surrounding region.
Maj. Gen. Vladimir Zolotukhin is quoted by Russian news agencies as saying they left Sunday through the town of Hamouria.
After a morning of calm, shelling and ground battles resumed across eastern Ghouta on Sunday afternoon, said the Britian-based Syrian Observatory for Human Rights, which is monitoring the conflict. The Observatory said 16 people were killed in the hospital including two pregnant women. The government offensive has pushed further into eastern Ghouta, chipping away at one of the largest and most significant opposition bastions since the early days of the rebellion - communities where some 400,000 people are estimated to be holed up. Turkey's military said it is combing the town for land mines, and tweeted a video showing Turkish soldiers in the town's center.
The Syrian army and allied forces have recaptured 70 per cent of the territory that was under insurgent control in the enclave, it said on Friday.
When Damascus condemned the Turkish offensive as a "violation of Syria's sovereignty, Turkey's foreign minister Mevlut Cavusoglu stated preservation of Syrian territorial integrity is the goal of both Ankara and Damascus and Turkish troops would not engage the Syrian army". Thousands of civilians have fled in recent days amid heavy airstrikes.
The Turkish military has pushed the YPG militia back from the border and almost encircled it with advances on the western and eastern flanks of Afrin town.