The Syrian Observatory for Human Rights said four couples and three children were killed in their homes in the centre of town as it came under heavy bombardment and Turkish air strikes aimed at clearing the way for a rebel advance.
The rebels gained ground, dodging snipers and deadly roadside bombs planted across the jihadist bastion.
With help from Turkish soldiers, artillery fire and air strikes, the "Euphrates Shield" rebel alliance launched a major push into the northern town.
The rebels began their offensive at midnight, seizing control of a hospital building, a key roundabout and a school complex in the town's western half, Abu Jaafar said.
Some rebels could be seen taking a short break on the western edges of the city after having spent around 12 hours in fierce street fighting.
"We've faced some difficulty with Daesh snipers -- there were more than 10 snipers last night that surprised our troops," Abu Jaafar said, using the Arabic acronym for IS.
He said roadside bombs that could be detonated remotely had been particularly deadly.
Al-Bab is IS's final stronghold in the northern Syrian province of Aleppo -- second only in importance to the jihadist group's de facto capital in neighbouring Raqa.
Turkey began military operations in Syria in August, crossing south into the war-ravaged country to fight both IS and Kurdish forces that it regards as terrorists.
Euphrates Shield forces initially swept across swathes of border territory easily, but they stalled around Al-Bab in December.
"There is fierce bombardment from Turkish forces on IS positions in Al-Bab," the Britain-based monitor said.
It has said that more than 110 civilians have been killed in just two weeks of Turkish shelling on the town, but Ankara has said it does its best to avoid civilian casualties.
Dozens of civilians have been fleeing Al-Bab every day, according to the monitor, leaving newly liberated areas as well as escaping territory still under IS control.
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