The government claims that it has regained control of three areas of Mogadishu - Tarbunka, Bakara and Howlwadag - since the battles began before dawn.
"We were surprised to see men in government uniforms fighting in Bakara," Halima Osman, a Mogadishu resident, told the Reuters news agency.
"They have recaptured four police stations between here and the palace, and they are advancing further."
But al-Shabab rejected the claims of military success.
"The enemy of Allah attacked our positions this morning and our fighters are defending themselves," Sheikh Ali Mahmoud Rage, a spokesman for al-Shabab, said.
"They have not not taken any positions from us."
Reporters confirmed that several bodies had been left on the ground and at least eight civilians were injured while trying to escape.
Sources at two Mogadishu hospitals told Reuters that they received a total of 85 wounded civilians on Friday.
Al-Shabab fighters, along with members of other armed opposition groups, have pledged to topple the government of Sheikh Sharif Sheikh Ahmed, the president.
Hassan Mahdi, a spokesman for Hizbul Islam, one of the other groups battling the government, said that troops had struck at their positions too.
|Scores of people have died as the fighting
has escalated in recent weeks [AFP]
"Al-Shabab and Hizbul Islam are counter-attacking ... we have pushed them back in some places. There are casualties, but I can't say how many. We are in the middle of fighting," he said.
The government on Friday acknowledged that it was unlikely that negotiations with fighters would prove fruitful.
"The opposition groups have been provoking us for the last three weeks," Mohamed Abdi Gandi, the Somali defence minister, said.
"We shall continue fighting this opposition with foreign ideologies. They want to destroy our government by the use of violence but it will not be."
In the past 10 days, more than 100 people have been killed and 46,000 have been displaced as the fighting has escalated.
Earlier this week an East African group called on the UN to impose an aerial and maritime blockade on Somalia to prevent the opposition fighters from acquiring more weapons.