As many as 50,000 people have fled separate offensives against rebel forces in northern and southern Syria in recent days, activists say.
Russian air strikes reportedly killed 31 people in the Eastern Ghouta outside Damascus on Friday, after 20,000 people left the region.
In the northern town of Afrin, where 30,000 people have fled, Turkish shelling killed at least 18 people.
Seven years of war have driven nearly 12 million Syrians from their homes.
At least 6.1 million are internally displaced while another 5.6 million have fled abroad.
More than 400,000 are believed to have been killed or are missing, presumed dead, since the uprising against President Bashar al-Assad began in March 2011.
The foreign ministers of Turkey, Russia and Iran – three countries closely involved in the conflict – have met in the Kazakh capital Astana to prepare for a summit on Syria in Istanbul next month.
How serious is the situation near Damascus?
Friday’s deaths in the Eastern Ghouta town of Kafr Batna were reported by the Syrian Observatory for Human Rights, a UK-based monitoring group.
Despite international calls for a ceasefire, there has been no let-up in the offensive and the Syrian army is now pushing into densely populated areas.
Residents have told BBC News there is intense street fighting between rebels and government soldiers in some parts of the enclave.
Humanitarian corridors controlled by the Syrian government reportedly allowed about 4,000 more people to leave the enclave on Friday despite the fighting.
According to the Observatory, nearly 20,000 civilians fled rebel-held areas in the region on Thursday.
The UN children’s agency Unicef is sending representatives to assess the needs of evacuees in temporary shelters in Damascus.
A Unicef spokeswoman said its current response plan could deal with up to 50,000 people.
Pro-government forces are believed to have recaptured 70% of the region after three weeks of intense fighting against rebels there.
The crushing of the rebel enclave outside Damascus would be a major victory for President Assad.
What is happening on the Turkish border?
Afrin, a town populated mainly by ethnic Kurds near the frontier, has been under bombardment from the air and the ground by Turkish forces and their local Syrian allies.
Some 30,000 people have fled the city and nearby villages, according to the Observatory. They headed towards villages held by Syrian government forces.
Turkey is targeting the Kurdish People’s Protection Units (YPG), a US-backed militia that it regards as an extension of the Kurdish rebels on its own territory.
Hundreds more families left overnight as shelling continued, the Observatory says. Five children were among those killed on Friday, it reports.
Brusk Hasakeh, a YPG spokesman, told Reuters news agency that Turkish forces and their Syrian militia allies were trying to storm Afrin from the north but the YPG and its women’s affiliate, the YPJ, were fighting back.
Turkish President Recep Tayyip Erdogan told a gathering in Ankara that his country would not stop until its mission to capture Afrin had been completed.
What are the latest diplomatic efforts?
While Turkey is opposed to President Assad, Russia and Iran are the Syrian leader’s closest allies.
Russian Foreign Minister Sergei Lavrov sought to portray the talks in Kazakhstan as an opportunity to bring lasting peace to Syria, saying “millions of Syrians are looking in the direction of Astana”.
Referring to the Eastern Ghouta, Turkish Foreign Minister Mevlut Cavusoglu said the bombing of civilians was unacceptable.
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