The main border crossing north of Syria's second city Aleppo remained closed, forcing huge crowds including women and children to sleep in tents or in the open.
"There are no longer enough places for families to sleep," said Ahmad al-Mohammad, a field worker with medical aid group Doctors Without Borders (MSF).
"Most of the families left with just the clothes they were in," he told AFP, adding that the cold and the crowded conditions were causing health problems including diarrhoea.
The United Nations says up to 31,000 people have fled Aleppo city and surrounding areas in recent days, as government forces backed by Russian warplanes press an offensive that could encircle the rebel-held part of the city.
"We are asking Turkey to open its border to all civilians from Syria who are fleeing danger and seeking international protection," said UN refugee agency (UNHCR) spokesman William Spindler.
EU president Donald Tusk said Russian air strikes were "making an already very bad situation even worse".
"As a direct consequence of the Russian military campaign, the murderous Assad regime is gaining ground, the moderate Syrian opposition is losing ground and thousands more refugees are fleeing towards Turkey and Europe."
Syria's nearly five-year-old conflict has claimed 260,000 lives and displaced half the population.
On Tuesday a suicide car bomb claimed by the Islamic State group killed at least nine people at a police club in the north of Damascus, according to the Syrian Observatory for Human Rights, a Britain-based monitor.
'They are trapped'
Turkey's Deputy Prime Minister Numan Kurtulmus said Monday that a "worst case scenario" could see up to 600,000 refugees arrive at the border.
"Our objective for now is to keep this wave of migrants on the other side of Turkey's borders as much as is possible, and to provide them with the necessary services there," Kurtulmus said.
Turkey already hosts 2.5 million Syrian refugees, but has come under pressure to allow in more, as well as to prevent them from seeking to reach Europe.
NATO said Tuesday it would take any request to help with the refugee crisis "very seriously," after Ankara and Germany said they would seek the alliance's help combatting people smugglers.
Turkey's Oncupinar border crossing north of Aleppo city remained closed Tuesday with only medical emergencies allowed through. A Turkish official said four wounded Syrians had been let in on Monday.
The UN's humanitarian aid agency OCHA said on Monday that eight informal camps on the Syrian side of the border were at "full capacity".
MSF said aid groups were distributing warm clothes and mattresses to those stranded on the Syrian side.
"They are trapped," Mohammad said. "They've left their homes and everything they have behind, and they can't get into Turkey."
The UN warned that 300,000 people in eastern Aleppo city could be cut off from humanitarian aid if government forces encircle the area.
Government sieges have been employed to devastating effect against other former rebel bastions.
A report from Washington-based The Syria Institute and PAX, a peace organisation based in the Netherlands, said Tuesday that more than one million Syrians are living under siege.
It said the crisis was "far worse" than acknowledged by the UN, which in January estimated the number at 486,700.
Strikes pound north Aleppo
The World Food Programme said it had begun food distributions to the displaced, despite the severing of access and supply routes in the region.
"We are making every effort to get enough food in place for all those in need," said WFP Syria country director Jakob Kern.
Syrian government forces backed by allied militias and Russian air strikes began a major operation in the northern province of Aleppo last week.
They are now around 20 kilometres (12 miles) from the Turkish frontier.
The regime advances came as peace talks in Geneva collapsed last week in part over rebel anger about the government offensive.
More than 21 suspected Russian air strikes hit targets in several towns northwest of Aleppo city and in the northern countryside on Tuesday, the Observatory said.
Suspected Russian air strikes also hit parts of Tal Rifaat during the night, according to the monitor, which relies on a network of sources on the ground.
Regime forces and their allies meanwhile were fighting rebels, including jihadists from Al-Qaeda-linked Al-Nusra Front, in the southwestern countryside of Aleppo province.
