The BBC's Ian Pannell said the jihadists were putting up a hard fight.
BBC producers Joan Soley and Peter Emmerson are also embedded with the Iraqi troops.
BBC Arabic's Feras Kilani caught in crossfire.
This picture from Feras Kilani shows the destruction of a neighbourhood on the outskirts of the city.
There are thought to be up to 1.5 million civilians in Mosul, trapped between IS and Iraqi forces.
The team were with the Counter-Terrorism Service (CTS) when it retook Bazwaya, about 3km (2 miles) east of Mosul, on Monday.

After his opening 66, two immediate birdies moved the 37-year-old to a share of the lead but six bogeys in windy conditions saw him losing ground.
A record nine players share the halfway lead on eight under.
These include first-day leader Welshman Bradley Dredge (72) and English Ryder Cup player Andy Sullivan (68).
Thailand's Kiradech Aphibarnrat, South African duo Thomas Aiken and Jaco Van Zyl, Spanish pair Jorge Campillo and Nacho Elvira, Finland's Mikko Korhonen and South Korea's Jeunghun Wang are also in the leading nine-strong group.
Ireland's Paul Dunne is sharing 10th place, a stroke off the pace, after adding a 70 to his opening 67.
2010 US Open champion McDowell has dropped to 89th in the world rankings, and as it stands, is not in the fields for either the WGC World Match Play in March or the Masters a month later.
As Iraqi special forces enter the city of Mosul for the first time since the start of the campaign to retake it from so-called Islamic State, a BBC team embedded with the troops is tweeting from the front line as the battle unfolds.
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Graeme McDowell dropped five off the pace at the Qatar Masters after struggling to a three-over-par 75 in Friday's second round in Doha.