About 1.1 million people have been displaced since Syrian rebels launched the offensive that ousted former president Bashar al-Assad, the UN’s humanitarian agency has said, as fighting between different factions continues.

“As of 12 December, 1.1 million people have been newly displaced across the country since the start of the escalation of hostilities on 27 November. The majority are women and children,” the UN Office for the Coordination of Humanitarian Affairs (OCHA) said in a statement.

Among those displaced are more than 100,000 people who have fled into Kurdish-administered areas in northern Syria amid escalating factional fighting and fears of retaliatory attacks.

Tensions appear to be concentrated primarily on the town of Manbij, north-east of Aleppo, and the mixed Arab and Kurdish town of Deir Ezzour in eastern Syria.

After the collapse of Assadist forces last week, Kurdish and Arab units fighting under the banner of the US-backed Syrian Democratic Forces (SDF) have advanced, in some places clashing with the Turkish-backed rebel groups of the Syrian National Army, in an effort to secure swaths of territory in northern and eastern Syria.

In a gesture of unity, the Kurdish-backed administrations of north-eastern Syria declared they would fly the flag of independence long used by opposition forces across the country, in order to “affirm the unity of Syria and its national identity”.

The SDF commander, Gen Mazloum Abdi, said US mediation had helped broker a ceasefire agreement in Manbij, but that his forces “continue to resist and stop the growing attacks from the west of the Euphrates”, as Turkish-backed rebel groups attempted to take control of the town. Despite the ceasefire, reports continued of fighting in the centre of Manbij.

“Our goal is to cease fire throughout Syria and enter into a political process for the future of the country,” said Abdi.

Hayat Tahrir al-Sham, the Islamist faction that is in control of much of Syria, has not clashed with Kurdish-led forces. Rebel forces in eastern Syria, however, drove Kurdish fighters out of Deir Ezzour amid confusion about who controls the town and rising fears about the presence of Islamic State (IS) fighters in the area.

Turkey, which regards the SDF and associated Kurdish fighters as terrorist groups, has also launched attacks on Kurdish troops. Ankara-backed forces struck a Kurdish convoy that it said was carrying heavy weapons looted from Syrian government arsenals.

The SDF said its forces were “repelling an attack” by Turkish-aligned forces at the Tishreen dam, near Manbij. “Fierce clashes continue amid fears for the dam,” it said, blaming bombardment by Turkish warplanes and tanks.

An estimated 900 US troops remain in eastern Syria to back Kurdish forces and other rebel factions battling to prevent a resurgence by IS.

Abdi told Sky news that his forces had been forced to pause the fighting against IS in eastern Syria due to mounting attacks by Turkish forces, prompting fears of an escape or prison break by jihadist militants held in camps in the east of the country.

IS “is now stronger in the Syrian desert. Previously, they were in remote areas and hiding, but now they have greater freedom of movement since they face no issues with other groups and are not engaged in conflict with them,” he told Sky News.

His forces have witnessed an increase in IS activities in areas under SDF control, he added, including killing several members of the SDF near Al Hasakah.

Speaking before his departure to Turkey on a stop in Jordan, US secretary of state Antony Blinken on Thursday acknowledged “real and clear interests” by Turkey about the PKK, the Kurdish fighters that Ankara links to the SDF.

“At the same time, again, we want to avoid sparking any kinds of additional conflicts inside of Syria,” Blinken told reporters in Aqaba, Jordan.

“And part of that also has to be ensuring that IS doesn’t rear its ugly head again. And critical to making sure that doesn’t happen is the so-called SDF, the Syrian Democratic Forces, that we’ve been supporting,” he said.

Blinken later told Turkish President Recep Tayyip Erdoğan that Syrian civilians need to be protected, during a meeting at Ankara airport, according to the state department.

Amid the chaos and fighting, rights groups warned that civilians were suffering the most.

“The situation is exacerbating an acute and longstanding crisis, with overcrowded camps and severely damaged infrastructure and a lack of water, power, healthcare, food and weather-appropriate shelter,” said Human Rights Watch.

The non-profit organisation also warned of widespread ill-treatment by Turkish-backed rebel groups in the area, including unlawful detentions, sexual violence and torture, land theft and extortion.

Adam Coogle, the deputy Middle East director at Human Rights Watch, said: “Amid the extraordinary events taking place in Syria, intense fighting and fear of retaliation and violence by armed groups is displacing thousands of civilians to areas unprepared for such an influx.”

Syria’s interim government vowed on Thursday to institute the “rule of law”. “All those who committed crimes against the Syrian people will be judged in accordance with the law,” the new government’s spokesperson, Obaida Arnaout, told AFP.

He also said that the country’s constitution and parliament would be suspended during a three-month transition.

“A judicial and human rights committee will be established to examine the constitution and then introduce amendments,” Arnaout said.