At least 30 people have been killed in fresh government air attacks on the rebel-held enclave of Eastern Ghouta, a war monitor reported.

The Syrian government attacks on Saturday also injured dozens more, the Syrian Observatory for Human Rights (SOHR) said, as civilians attempted to flee the front lines in the besieged enclave.

The attack comes as thousands of civilians over the past days fled Eastern Ghouta, which has been under a month-long bombardment campaign by the government.

The UK-based SOHR said at least 10,000 people fled the Damascus suburb on Saturday, as deadly air raids continued.

On Friday, 46 civilians, including six children, were killed in the district of Kafr Batna, according to the Observatory.

Earlier on Friday, it was reported that between 12,000 and 13,000 people had fled the area east of Damascus.

Abandoning their homes with whatever they could carry - some with their vehicles and others on foot - desperate civilians fled to areas under government control.

The humanitarian situation in the suburb east of Damascus has worsened since February 18, when Russian-backed Syrian warplanes intensified their attack on the enclave, killing more than 1,250 people in a relentless bombardment, despite efforts to implement a lasting ceasefire.

In an earlier attempt to drive armed opposition groups out, Syrian President Bashar al-Assad's government imposed a tight siege on Eastern Ghouta back in 2013 - shortly after it fell under rebel control.

As a result of the blockade, food shortages and a lack of medical supplies has left some 400,000 residents desperate for aid.

Assad's forces are inching closer to capturing the rest of the enclave, forcing civilians to flee. They have already split the enclave into three sections.

Among the thousands of civilians who fled, many were from the town of Hamouriyah, which has been at the centre of the fighting between rebels and military forces.

Government forces now surround Hamouriyah, with residents reporting a lack of access to water and medicine. But rebels claim they have retaken the district.

The United Nations refugee agency in Syria said people in Eastern Ghouta are continuing to leave the area on Saturday, with an estimated 12,000 to 16,000 people having left in the past days.

"The majority of those leaving are from the Hamouriyah area, but also include some medical cases from Douma," Mysa Khalaf, UNHCR's spokesperson, told Al Jazeera from Damascus.

"The actual number of people who have exited eastern Ghouta is not known, nor are the destinations of all evacuees," she said.

The UNHCR office in Syria's capital confirmed that there are three shelters in the Damascus area where people from Eastern Ghouta are arriving: Dwier, Hejelleh and Adra.

"The overall humanitarian scene is dire. The UN has delivered emergency items such as food, mattresses, blankets and hygiene kits and is preparing for further assistance to be delivered in response to the growing numbers of people arriving at the shelters," Khalaf said, describing the scenes at the shelters.

"Mobile clinics are providing medical services and vaccinations to children under five. Those in need of medical care are being treated at hospitals in Damascus."

Meanwhile, about 25 trucks of food aid were allowed into Eastern Ghouta's Douma district, according to the International Committee of the Red Cross (ICRC).

It is unclear how long the food supply would last in an area believed to be populated with as many as 125,000.

Thursday marked seven years since the Syrian civil war began, with more than 465,000 Syrians killed and millions more displaced.