The Britain-based monitor said more than 20,000 people had fled to western neighbourhoods of the city held by the government, with another 30,000 moving to areas held by Kurdish forces.
Syrian troops have seized at least a third of eastern Aleppo since renewing their bid to recapture all of the battered second city just over two weeks ago.
The former rebel stronghold has been under a government siege for more than four months, with international aid rations exhausted and other food stocks dwindling.
Many have chosen to go from the east to neighbourhoods held by Kurdish forces, which are officially aligned with neither the regime nor rebels, such as Sheikh Maqsud in the city's north.
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Once Syria's economic powerhouse, Aleppo has been ravaged by the conflict that began with anti-government protests in March 2011.
The rebel-held east, which fell from government control in 2012, has been particularly savaged by the conflict, with widespread destruction caused by repeated regime attacks.
The loss of east Aleppo would be potentially the worst blow for rebel forces since the conflict began.
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