The first convoy of humanitarian assistance for victims of Monday’s earthquake has crossed into north-west Syria, as the death toll in Turkey and Syria climbed to more than 21,000 amid fading hopes of finding survivors under rubble in freezing weather.

Six trucks passed through the Bab al-Hawa border crossing from Turkey carrying tents and hygiene products, as Turkey said it was working to open two more border crossings with Syria to allow in more humanitarian aid.

The World Health Organization (WHO) warned that without quick help, tens of thousands of people living in the open could be under threat from the harsh conditions, and with the death toll likely to rise significantly higher.

Dr Wassel Ajerk, a general surgeon at Idlib city hospital and project manager for the Syrian Expatriate Medical Association, said the situation in rebel-held north-west Syria was “miserable”. He said it faced the same situation as Turkey but without any government support.

Robert Holden, the WHO’s incident response manager, said in Geneva that many people were surviving “out in the open in worsening and horrific conditions”, with water, fuel and electricity supplies severely disrupted.

“We are in real danger of seeing a secondary disaster which may cause harm to more people than the initial disaster if we don’t move with the same pace and intensity as we are doing on the search and rescue side,” he said. “People need the basic elements to survive the next period.”

In Turkey, a survivor in the city of Antakya called for the government to evacuate people from devastated areas. “Especially in this cold, it is not possible to live here,” said Ahmet Tokgoz. “If people haven’t died from being stuck under the rubble, they’ll die from the cold.”

Just one border crossing, in Turkey’s southern Hatay province, is open for transporting life-saving aid to rebel-held regions of Syria under the authorisation of the UN security council. Humanitarian aid for rebel-held areas usually arrives through Turkey via a cross-border mechanism created in 2014 by a security council resolution.

Turkey’s foreign minister, Mevlüt Çavuşoğlu, said late on Wednesday: “There is damage to some roads on the Syrian side of the border. There has been some difficulty for our and international aid to get through because of the destruction. For this reason, we are working on the opening of two more posts.”