Lebanese officials estimate that more than 915,000 people or approximately a quarter of the population has been made homeless by Israel's four week offensive.
One aid agency said it was now "almost impossible" to work outside Beirut.
Many truck drivers are refusing to deliver aid for fear of being targeted by Israel.
Israel has warned that it will strike any vehicle moving south of the Litani River, an area that includes the port city of Tyre.
Much of Tyre now resembles a ghost town.
Hospitals say they are just days away from running out of the fuel needed to power emergency rooms and life-support machines.
British-based aid agency Oxfam warned of a "logistical nightmare" in reaching the hardest-hit victims.
It said it was "almost impossible" to operate beyond the capital Beirut.
"There are attacks on any movement," said Ian Bray, Oxfam's spokesman on humanitarian emergencies.
He added that hospitals in southern Lebanon were running out of fuel and needed deliveries.
"We are getting reports that truck drivers are refusing to work because they are afraid of being hit by Israeli missiles,” he said.
The European Commission's Humanitarian Office (ECHO) in Brussels said that even if the Israelis agreed to let aid convoys through, it was no longer feasible to make large deliveries.
"Because the roads have been bombed, big trucks can't get through," ECHO's media relations officer Simon Horner said.
"Even if you get agreement for a convoy, aid packages have to be loaded into little vans which can use minor roads.
"If the conflict goes on there will be a humanitarian catastrophe; it's already a crisis and it could become a catastrophe."
Medecins Sans Frontieres (Doctors Without Borders) said Israeli bombing of bridges was a "huge blow for delivery of vital supplies to people trapped in southern Lebanon."
It said four tonnes of supplies were carried by hand for half a kilometre along a human chain, with journalists at the scene lending a hand.
A tree trunk was used as a makeshift bridge at Qasimiya, 11 km north of Tyre, after Israeli warplanes destroyed the two bridges on the coast road.
"The bombing of the Qasimiya crossing is yet another major blow in our endeavour to provide desperately needed supplies to hard-hit populations in southern Lebanon," said Christopher Stokes, MSF coordinator in Lebanon.
The Beirut authorities say hospitals treating thousands of wounded in Lebanon will run out of fuel within days unless Israel relaxes its sea blockade.
The situation was "worsening" though the United Nations was working to facilitate the passage of two fuel ships berthed in Cyprus, UN officials said.
Just one of their convoys had managed to move south of Beirut on Wednesday.