The refugees are coming from Gaza's north, fleeing Israel's bombing campaign. They stream down the Salah al-Din road, which connects north to south, many thousands on foot, some with a few possessions but most bearing only their children and the clothes on their backs.

Tens of thousands have stopped in Deir al-Balah, a central Gazan city in the supposed safe zone which has been plunged into crisis by the influx. The refugees in Deir al-Balah are crammed into school buildings hastily repurposed as UN shelters, up to 70 people in a single classroom, surrounded by food waste and swarmed by flies.

"If you want to speak about space, we sleep on our sides because there is not even enough room to lie on our backs," said Hassan Abu Rashed, a 29-year-old blacksmith who fled with his family from Jabalia in Gaza City.

At the gate of one school in Deir al-Balah, Khaled Filfel, a 42-year-old father, was alone and stressed over a very specific need. "My 21-year-old daughter is disabled and I cannot get any nappies for her," he said. On top of that, he said, the pair had not been able to find drinking water or food so far that day.

There were two saving graces for Filfel, though. The first was that his wife and six other children happened to be out of Gaza when Hamas attacked Israel. The second was that someone had seen his daughter at the school that morning and offered them a room in a family home nearby. "Because of my daughter's condition they offered us shelter," Filfel said. "Some people here are looking out for each other."

Before the beginning of this war, the UN's refugee agency, Unwra, had contingency planning in place to house 1,500 displaced people in each school, the agency's Gaza director, Thomas White, told the BBC. The average school-turned-shelter is now housing 6,000 people - a total of 670,000 people across 94 shelters in the south.

"We have been overwhelmed by the numbers," White said. "There are people everywhere. The sanitation is overwhelmed, we're averaging about 125 people per toilet, about 700 per shower unit. You can feel the humidity of so many people crammed into these schools, you can smell the mass of humanity."

To escape the teeming classrooms and courtyards at the school in Deir al-Balah, some of its new residents take the short walk down to the beach front and spend the daylight hours there.

On Saturday morning, a young family was washing themselves and their clothes in the sea, trying to avoid the rubbish floating on the water and strewn on the sand. When they were done, they hung their clothes up under the sun. They had been in Deir al-Balah for three weeks.

"You could say that we have gone back to the dark ages," said the father, Mahmoud al-Motawag, 30. "We use the sea for everything," he said. "To wash ourselves, to wash our clothes, to clean our kitchen utensils, and now to drink when we cannot find clean water. We eat just one meal each per day, and we beg the fishermen to give us one or two fishes for the children."

Mahmoud, a farm worker from Jabalia, said his family had fled the bombing. He was sitting next to his two children, a boy and girl aged four and two, and his wife Duaa. The family spent all day at the beach, Mahmoud said, partly to wait for their clothes to dry but mostly to avoid for as long as they could returning to the baking hot tent on the school grounds that had become their temporary home along with 50 others.

As he spoke, Duaa, aged just 20, rested a hand on her large baby bump. She was due to give birth in a month, she said. With the local hospital already on its knees, she wondered if she might be forced to deliver at the dirty, overcrowded school.

"I am afraid," Duaa said. "I am afraid that the birth will take a long time, I am afraid for my baby, I am afraid that there will be no clothes or blankets. Everything was planned for the birth, and then everything changed."

For now, there was the daily strain of simply being a refugee while eight months pregnant. "I have this physical and mental fatigue," Duaa said. "My children are small and we have to stand in a queue for the toilets for 15 to 30 minutes. I have pain from washing and sitting for a long time by the sea. It doesn't go away."

Even if Duaa could reach the hospital in Deir al-Balah, it would not be a guarantee of a safe and comfortable birth. The Al-Aqsa hospital, like others across the Gaza Strip, is on its knees. As the refugees move south, so has the Israeli bombing, levelling buildings in residential areas of Deir al-Balah and sending dozens of badly wounded there.

"They are coming now, hundreds of injured people, dozens have injuries in the head and limbs," he shouted, over sounds of chaos in the background. "This is a massacre of our people."

Al-Duqran apologised and hung up. Later, when the chaos had died down, he called back, sounding shattered. "This is the hardest war that I have seen in my 20 years," he said. "Every day the wounded and the dead arrive by the dozens or hundreds. Children come with amputated limbs, upper and lower. They have severe head wounds."

Like other hospitals across Gaza, the Al-Aqsa was running low on almost everything it needed to function. "We are making beds from wooden pallets, we are missing nearly 90% of medicines," Al-Duqran said. "Everything from operation room trays to fraction fixing devices have run out, and in the ICU we will lose patients soon because we can simply no longer keep them alive."

As Israel's air and land attack on northern Gaza intensifies, people continue to flow down the Salah al-Din road into Deir al-Balah and all cities of central and southern Gaza.

But at many of the school shelters, there is no longer room. So refugees are building ramshackle lean-tos against the sides of the buildings, keen to be positioned as close as possible to a UN flag in the hope of protection from an air strike, but open to the elements as the weather worsens.

"People are living increasingly out in the open," said Thomas White, the Unrwa Gaza director. "Right now its remarkably warm for November, but by Wednesday we are expecting the cold weather to come through," he said. "People are going to be seriously exposed."

Every shop that was providing food to Gazans under a World Food Programme assistance scheme ran out of basic supplies on Friday, WFP spokeswoman Alia Zaki told the BBC. Bakeries have no gas to make bread, she said, and there was a potential wave of malnutrition in the making in Deir al-Balah and across Gaza.

"People are not eating enough to be healthy so their immune systems are weakened," Zaki said. "They are queuing for five or six hours for bread and coming back empty handed."

At the beach in Deir al-Balah on Saturday, this was the unwelcome prospect facing Mahmoud and Duaa. They were preparing mentally to leave the relative haven of the waterfront to go in search of bread.

"We could be waiting many hours, only to find the bakeries are closed again and we will have nothing again for our children," Mahmoud said.

"Our ancestors' lives were war and our lives have been war," he said, wearily. "And now the war has caught up with our children, too."