Iraqi special forces have launched an assault on one of the Islamic State of Iraq and the Levant (ISIL) group's strongholds, Fallujah, facing stiff resistance with the group launching suicide attacks.

The assault was launched in the early hours of Monday morning.

"Iraqi forces entered Fallujah under air cover from the international coalition, the Iraqi air force and army aviation, and supported by artillery and tanks," said Lieutenant-General Abdelwahab al-Saadi, the commander of the operation.

Al Jazeera's Omar Al Saleh:

The battle for Fallujah Iraqi army commanders estimate it will take them about 48 hours to clear Fallujah. I think this is very optimistic. There are between 400 and 1,000 ISIL fighters stationed inside the city - including its most experienced. In the early hours of Monday morning, Iraqi forces began efforts to advance from three fronts - mainly from the south and northeast. Fighting is going to be tough for both sides. We understand that there is heavy air power provided by the US-led coalition and the Iraqi air force. Fallujah is very symbolic for the government. It's very close to the capital, Baghdad, which can be reached in a 30-minute drive. Fallujah was the first Iraqi city to fall to ISIL in 2014. It was also the main Sunni city that fought against the Americans when they occupied it in 2003.

"CTS [Counter Terrorism Service], the Anbar police and the Iraqi army, at around 4am [01:00 GMT], started moving into Fallujah from three directions," he said.

"There is resistance from Daesh," he added, using an Arabic acronym for ISIL.

Al Jazeera's Omar Al Saleh, reporting from Erbil, cited military sources saying that at least 10 Iraqi security forces and members of allied Shia militias were killed in the early hours of the offensive, while 25 more were injured.

Also on Monday in Ramadi, which is less than 100km from Fallujah, Iraqi police said at least 15 special force soldiers were killed in an ISIL attack.

Meanwhile, at least nine people were killed and 26 were wounded in bombings north and northeast of the capital, Baghdad.

Fighting on Monday followed battles a day earlier, adding to the exodus of thousands of desperate civilians from the surrounding areas and deep concern for the many more trapped in the battlegrounds.

The week-old operation to capture Fallujah has so far focused on retaking villages and rural areas close to the central city, which lies just 50km west of Baghdad.

CTS's involvement will mark the start of a phase of urban combat in a Sunni city where US forces in 2004 fought some of their toughest battles since the Vietnam War.

Only a few hundred families have managed to slip out of the Fallujah area, with an estimated 50,000 people still trapped inside the city proper.

According to the Norwegian Refugee Council (NRC), around 3,000 people have managed to escape the Fallujah area since May 21.

A larger influx could be triggered when the urban battle between CTS and ISIL begins in earnest.

"Our resources in the camps are now very strained, and with many more expected to flee we might not be able to provide enough drinking water for everyone," said Nasr Muflahi, NRC's Iraq director.

"We expect bigger waves of displacement the fiercer the fighting gets."

$48 for a kilo of rice

The Fallujah operation has come at a human cost, rights groups said, amid battles between ISIL (also known as ISIS) fighters and the advancing Iraqi army and allied Shia militia.

One Fallujah resident told Al Jazeera by phone that there is lack of medicine and fuel in the city.

"There is some food. We have vegetables, enough to survive. But there is no rice and sugar, the price for a kilo of rice here reached $48," the resident said. "ISIL is on alert on the outskirts of the city. Its fighters have set up checkpoints and prepared ambushes, which prevent people form leaving."

The battle for Fallujah has raised fears of a sectarian backlash, as Shia militias have been accused of retaliation against the civilian population in areas retaken from ISIL.

Middle East analyst Michael Pregent, at the Hudson Institute in Washington DC, said the heavy involvement of Shia militias "is not going to win over the Sunni population that is needed to fight ISIS”.

"We are leaving Shia militia men lead by [Iranian General] Qassem Soleimani lieutenants to decide who is ISIS and who is a collaborator," he told Al Jazeera.

"If you look at the public relations photos coming out Fallujah you have women and children being given water bottles by the Shia militia men but where are the men? What they do is they take the men and put them somewhere else and the militia men decide who is a Daesh [ISIL] and who is a collaborator, and the punishment is the same."

Ranj Alaaldin, a Middle East analyst at the London School of Economics, said ISIL has significant local support in Fallujah.

"Those fighting the Iraqi forces in Fallujah are mostly local indigenous actors - tribes, militant groups that have had more than decades worth of mobilisation experience. It is fighters who took up arms against the US and then after that against the Iraqi government.

"Do not forget this is a city which has been resisting any control from the outside, be it forces of the US or the Iraqi government itself."