Dozens of fighters from the Islamic State of Iraq and the Levant (ISIL) and Syrian government forces have been killed in ongoing battles in Hasakah province, a monitoring group and activists said.
The fighting has displaced more than 100,000 people, the UN said.
At least nine ISIL fighters and 12 government soldiers were killed during clashes on Monday, the UK-based Syrian Observatory for Human Rights reported.
On the eastern side of Hasakah city, Kurdish People's units (YPG) foiled an ISIL suicide car bomb in the neighbourhood of Ghweran and captured three villages from ISIL fighters, activists and the Observatory told Al Jazeera.
ISIL launched its offensive on Syria's northeastern province of Hasakah, which borders Turkey, on June 25.
Since then, at least 71 government soldiers and 48 ISIL fighters have been killed.
The Observatory said more than 30 civilians have been killed in the clashes, with an increase in the death toll expected.
ISIL moved closer to Hasakah city last month, but the city remains under opposition and YPG control.
Government air strikes have targeted ISIL fighters in several neighbourhoods in and around Hasakah since Thursday.
About 2,000 people are trapped in the neighbourhoods of al-Nashwa and al-Sharia due to the fighting and government air strikes, the UN reported over the weekend.
The UN also said at least 120,000 have been displaced due to the fighting within Hasakah city and its surrounding villages.
The population of Hasakah province in 2011 was 1.5 million - with 300,000 living in Hasakah city, which has a mixed Arab, Kurdish, and Christian population.
The UN also said they expect more people will try to flee in the next few days.
Sunday marked one year since ISIL's leader, Abu Bakr al-Baghdadi, declared himself a caliph of what he called the "Islamic State" in Iraq and Syria.
|ISIL fighters take part in a military parade along the streets of northern Raqqa province, June 30, 2014 [Reuters]
ISIL now controls almost 50 percent of Syrian territory, declaring eight "Islamic states" inside the country.
They currently have presence in nine out 14 Syrian provinces and control many significant oil and gas fields.
Since Baghdadi's statement a year ago, the Observatory said it has documented 3,027 executions carried out by ISIL, including those of 1,787 civilians, 74 of them children.
More than half of those executed were civilians and more than half of the executed civilians were members of the Sunni Shaitat tribe, which revolted against ISIL south of Deir Ezzor city in August 2014.
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The overall toll includes the mass killings that took place in the surprise ISIL attack on Kobane last week after being forced out in January. Activists told Al Jazeera almost 300 people were killed in the attack.
The Observatory also reported that at least 8,000 ISIL fighters have been killed in clashes with Syrian rebels and YPG Kurdish forces, and in US-led air strikes that started in September 2014.
Source: Al Jazeera