* Heavy explosions, gunfire on southern fringes of Falluja

* Army presses assault to retake IS bastion near Baghdad

* Seeking to help protect capital from suicide attacks

* Over 20 people killed in latest Baghdad bombings

By Saif Hameed and Kareem Raheem

BAGHDAD, May 30 (Reuters) - The Iraqi army stormed to the southern edge of Falluja under U.S. air support on Monday, launching a direct assault to retake the city from Islamic State (IS) militants and help protect the nearby capital Baghdad from suicide bombings.

As government forces pressed their onslaught, a car bomb as well as suicide bombers driving a car and a motorcycle killed more than 20 people and injured over 50 in three districts of Baghdad, police and medical sources said.

Bolstered by Iranian-backed Shi'ite Muslim militia, the Iraqi army launched its operation to recover Falluja on May 23, first by tightening a six-month-old siege around the city 50 km (30 miles) west of Baghdad.

Falluja in January 2014 became the first Iraqi city to fall to the ultra-radical Sunni Muslim IS, and it subsequently overran wide areas of the north and west of Iraq, declaring a caliphate that included seized territory in neighbouring Syria.

On Monday, army units advanced to the southern entrance to Falluja, "steadily advancing" under air cover from the U.S.-led coalition, according to a military statement read out on state TV. A Reuters TV crew on the scene said explosions and gunfire were ripping through Falluja's southern Naimiya district.

A Shi'ite militia coalition known as Popular Mobilization, or Hashid Shaabi, were seeking to consolidate the siege by dislodging militants from Saqlawiya, a village just to the north of Falluja.

The militias have pledged not to take part in the assault on the mainly Sunni Muslim city itself to avoid aggravating sectarian strife.

 Iraqi army storms to edge of Islamic State-held Falluja


ISLAMIST MILITANT STRONGHOLD

Falluja is a bastion of the Sunni insurgency that fought the U.S. occupation of Iraq and the Shi'ite-led Baghdad government that took over after the fall of Iraqi strongman Saddam Hussein, a Sunni, in 2003.

The offensive is causing alarm among international aid organization over the humanitarian situation in the city, where more than 50,000 civilians remain trapped with limited access to water, food and health care.

Falluja is the second-largest Iraqi city still under control of the militants, after Mosul, their de facto capital in the far north that had a pre-war population of about 2 million.

Kurdish Peshmerga forces on Sunday launched an attack to oust Islamist militants from a handful of villages about 20 km (13 miles) east of Mosul so as to increase the pressure on Islamic State and pave the way for storming the city.

Prime Minister Haider al-Abadi hopes to recapture Mosul later this year to deal a decisive defeat to Islamic State.

Abadi announced the onslaught on Falluja on May 22 after a spate of bombings that killed more than 150 people in one week in Baghdad, the worst death toll so far this year.

Monday's bombings targeted two densely populated Shi'ite districts, Shaab and Sadr City, and one predominantly Sunni suburb, Tarmiya, north of Baghdad.

A car bomb in Shaab killed 12 people and injured more than 20, while in Tarmiya seven were killed and about 20 injured by a suicide bomber who pulled up in a car outside a government building guarded by police. In Sadr City, a suicide bomber on a motorcycle killed two people and injured seven.

(Writing by Maher Chmaytelli; Editing by Mark Heinrich) ((maher.chmaytelli@thomsonreuters.com; +9647901917030;))