SOHR, The Syrian Observatory for Human Rights said more than 200 people had been killed every day for the past three weeks.
Both government forces and rebels were trying to gain territory, it said.
This was possibly to strengthen their negotiating position at the peace talks, it added.
The talks between the Syrian government and opposition groups are continuing in Switzerland, but there is no sign of a breakthrough.
At least 4,959 people had died in the three-week period since 22 January, the SOHR reported. About a third of the casualties were civilians, including 515 women and children.
That casualty rate was higher, the group added, than in any other three-week period since the conflict began in March 2011.
The claim by the British-based group – which has links to the opposition – has not been independently verified.
On Wednesday, the UN-backed evacuation of civilians and delivery of aid continued in the besieged rebel-held Old City area of Homs.
More than 200 civilians left, joining hundreds allowed out since a truce was agreed on Friday.
But concerns remain over the fate of men of military age who are being held by the Syrian authorities after attempting to leave.
Also on Wednesday, Syrian government troops – backed by their Lebanese Shia ally Hezbollah – stepped up an assault on the strategic opposition-held town of Yabroud, near the Lebanese border, with at least 13 air strikes reported.
Lebanese officials told the BBC dozens of families were streaming over the border in anticipation of a major offensive.
Meanwhile, sharp words were exchanged by Washington and Moscow after Russia again objected to a draft UN Security Council resolution – this time, one that would call on all sides to allow aid workers access across Syria.
That was criticised by US President Barack Obama, who called Russia a “holdout” and suggested that by blocking the resolution it too was responsible for “starving civilians”, along with the Syrian government.
A Russian foreign ministry spokesman dismissed the criticism as a “biased distortion”, highlighting Russia’s role in helping achieve the ceasefire in Homs and insisting Russia was as concerned about the humanitarian situation in Syria as Washington.
But the BBC’s Nick Bryant at the UN in New York says that although Russia has rejected this present draft, it has not slammed the door on an alternative resolution.
In Geneva, Syrian government and opposition negotiators met face-to-face again on Wednesday.
The opposition proposed a transitional governing body (TGB) be set up to oversee a UN-monitored ceasefire across Syria and steer it out of the war, which has lasted nearly three years.
It would expel all foreign fighters, allow full humanitarian access and achieve a political solution to the war.
But regime representatives refused to discuss the plan, saying it was broaching political questions prematurely, reported AFP news agency.
UN envoy Lakhdar Brahimi has lamented a lack of progress in the talks – bringing forward a meeting with Russian and US officials by one day to Thursday in an apparent attempt to get them to apply pressure to the opposed Syrian sides.
The first round of talks ended last month with no firm agreements and both sides trading insults.
President Bashar al-Assad’s government has ruled out any transfer of power.
The civil conflict has claimed more than 136,000 lives and has driven 9.5 million people from their homes.
Top photo: Dozens of Syrian families from Yabroud have streaming over the Lebanese border in anticipation of a major offensive. Lebanon is bracing for a flood of Syrian refugees.
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