The government declared victory in the city west of Baghdad last week after IS had held it for more than two years.
But the United States-led coalition says only one-third of the city has been cleared of militants while the rest of the city remains contested.
Just four days ago, Iraqi prime minister Haider al-Abadi himself declared victory over I-S militants in the city of Fallujah.
"We promised to liberate Fallujah, and we took it back. Our brave forces went into the town of Fallujah and took control of areas inside the city."
While he was quick to proclaim victory over IS, also known as ISIL, he did acknowledge the fight was not over.
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"There are still some pockets that need to be cleared."
In fact, United States military officials say, only one-third of the city has been cleared of militants.
An Al-Jazeera online report quotes US Department of Defence spokesman Matthew Allen as saying many other parts of the city remain in dispute.
"The rest of the city remains contested, and we will continue to support the government of Iraq in eliminating ISIL from Fallujah. (It) has been and will continue to be a difficult fight."
In recent days, there have been persistent clashes between Iraqi forces and IS fighters holed up in dense residential neighbourhoods along the city's edge.
US officials say that is where the resistance is strongest.
But Iraq's Counter Terrorism Service says it is close to finishing up the final stages of purging the city of all IS militants.
Staff General Abdul Wahab Alsaidi says the fighting is ongoing in the western part of the city but Fallujah should be finished soon.
"They try to attack our troops to prove their existence. Their greatest effort is on the western axis. When the western axis collapses and the troops reach the Jolan neighborhood, then you'll not hear any other gunshot in Fallujah."
The operation has fuelled an exodus of thousands of desperate families.
Many have been forced to sleep in the open desert and at makeshift camps on the city's outskirts, overwhelming aid agencies.
The Norwegian Refugee Council has told Al Jazeera the city was already enduring difficult humanitarian conditions.
It says many civilians have been forced to eat rotten dates and animal feed and drink from a river that is undrinkable.
Among the fleeing civilians is a woman named Srud, who says she witnessed beatings, executions and terror while under IS control.
She says she managed to escape with her husband and three children.
"When we started running, they came after us, the IS militants were shooting at us with guns from behind. But we had decided we would rather die on the road with pride than in the hands of the IS."
Some reports say more than 83,000 civilians have fled the city since the operation began a month ago.
In total, the United Nations says, more than 3.4 million Iraqis have fled their homes across northern and western Iraq since the city fell to IS back in January 2014.
