The advance on several fronts is part of an offensive to rout militants from Fallujah and recapture the city west of Baghdad, which has been held by IS for more than two years.
IS fighters were said to be putting up fierce resistance and launching counter-attacks as the operation continued.
The offensive on Fallujah, backed by paramilitary troops and aerial support from the US-led coalition, was first launched about a week ago.
The battle for the strategic city is likely to be a protracted one, with Iraqi forces advancing slowly to minimise civilian casualties. Tens of thousands of civilians are believed to be still inside the city, trapped by the fighting.
The deadliest of Monday’s IS bombings took place in the northern, Shiite-dominated Shaab neighbourhood of Baghdad where a suicide bomber rammed his explosives-laden car into a checkpoint next to a commercial area, killing eight civilians and three soldiers. The explosion also wounded up to 14 people, a police officer said.
A suicide car bomber struck an outdoor market in the town of Tarmiyah, about 31 miles north of Baghdad, killing seven civilians and three policemen, another police officer said, adding that 24 people were wounded in that bombing.
In Baghdad’s eastern Shiite Sadr City district, a bomb motorcycle went off at a market, killing three and wounding ten, police said.
In an online statement, IS claimed responsibility for the attacks, saying they targeted members of the Shiite militias and a government office. Since launching the Fallujah offensive, Iraqi government troops have mostly been fighting IS in the outskirts of the city to tighten the siege.
By Sunday, the troops had recaptured 80 percent of the territory around Fallujah, according to Iraqi Major Dhia Thamir.
At dawn on Monday, Iraq’s elite counter-terrorism forces started pushing into Fallujah from its southern edge.
“This is the decisive battle for us and for Daesh,” said General Saad Harbiya head of Fallujah operations for the Iraqi army. He described the clashes as “fierce,” with IS deploying snipers and releasing a volley of mortar rounds on the Iraqi forces.
Fallujah is one of the last major IS strongholds in western Iraq.
The extremist group still controls territory in the country’s north and west as well as Mosul, Iraq’s second largest city.
The 500-700 IS fighters holed up in Fallujah are expected to be some of the group’s best-trained, a special forces commander at the scene said.
In a televised speech to parliament on Sunday, Iraqi prime minister Haider al-Abadi called on Fallujah residents to either leave the city or stay indoors.
Government officials and aid groups estimate that more than 50,000 people remain inside the centre of the Sunni majority city.
Humanitarian groups say that as the violence intensifies, their concerns for civilians trapped inside Fallujah mount. “With every moment that passes, their need for safe exits becomes more critical,” said Nasr Muflahi, of international humanitarian group the Norwegian Refugee Council.