The number of civilians displaced since last month's start of the offensive against Mosul -- the Islamic State group's last stronghold in Iraq -- rose to 68,000, but most of the city's population remained trapped.
A coalition aircraft carried out an air strike on Mosul's "third bridge", leaving a British-era bridge in the centre as the last crossing of the Tigris River running through the city.
Coalition spokesman Colonel John Dorrian said IS fighters has been using the bridges to re-supply the eastern side of the city, "essentially rotating their forces".
"We're not going to let that happen," he told AFP.
A member of the provincial council for Nineveh, of which Mosul is the capital, said the central bridge was the last remaining after four others over the Tigris had been destroyed.
IS has put up fierce resistance to defend Mosul, the city where its leader Abu Bakr al-Baghdadi proclaimed a "caliphate" in June 2014.
The eastern bank of the Tigris was expected to offer less resistance when tens of thousands of Iraqi forces launched the offensive on October 17 to retake the city.
Most of the jihadists' traditional bastions are on the western side of Mosul, as is the old city whose narrow streets will be hard to penetrate for the government forces' armoured vehicles.
A spokesman for the International Organization for Migration voiced concern that the lack of bridges could further trap civilians, who IS have routinely used as human shields.
"That would deprive a lot of families with an avenue to get away from the fighting," Joel Millman told AFP.
Over five weeks, Iraqi forces advancing on several fronts have made considerable progress in the advance, but the fighting in Mosul itself has been tough.
Elite forces from the Counter-Terrorism Service (CTS) -- the best trained units in the country -- have faced a daily barrage of mortar and sniper fire.
IS fighters have also launched car bombs driven by a seemingly endless supply of suicide bombers against their positions and convoys in the city.
"We're just in the very toughest part of the fight," Dorrian said.
Iraqi forces are "up against an enemy where the most likely course of action and the most dangerous course of action are often one and the same," the coalition spokesman said.
The intensity of the fighting has made it difficult for the hundreds of thousands of civilians in Mosul to flee to safety in camps erected around the city.
No safe routes
OCHA said the aid response to the offensive was growing in complexity, with varying needs for different categories of civilians.
"Humanitarian needs are severe among displaced families in and out of camps, vulnerable residents of retaken communities, and people fleeing the intense fighting in Mosul city," it said.
The UN had initially predicted that 200,000 civilians could be forced from their homes in the first few weeks of the offensive, Iraq's biggest military operation in years.
Iraqi forces have so far been sending the message to the population of Mosul that they should stay at home and not try to flee through the front lines.
Many residents of Mosul have indeed hunkered down homes as Iraqi forces took on IS fighters in fierce street battles.
That has restricted both the government forces' ability to use heavier weaponry against the jihadists and aid groups' ability to deliver assistance to civilians in need.
"While Mosul is under ongoing heavy attack, there are currently no safe routes out of the city," Norwegian Refugee Council (NRC) spokeswoman Becky Bakr Abdulla told AFP.
"Civilians are facing an extremely difficult decision of either staying in their homes stuck in the crossfire or risk their lives in an attempt to find their way out of the city," she said.