One of the Iraqi civilians who risked an escape from the sprawling battle for Fallujah made it as far as the Euphrates River. He was there for all to see on Sunday morning: His body, tied to the side of a boat, bobbed in the muddy waters next to a rickety bridge that separates Baghdad from the violence of Anbar Province. "Sheikh, sheikh, see this man! He drowned," said a young boy, pointing, as he approached the window of a truck that was slowly crossing the bridge, carrying medicine. "See, see his body." The thousands of civilians who managed to flee Fallujah and its outskirts and make it to government-controlled areas in recent days faced harrowing journeys, often at night and under fire from Daesh (so-called IS) militants who had been trying to use them as human shields. Many crossed the wide Euphrates in makeshift boats, and local officials said more than a dozen drowned in the last few days, dying in their own country in the same way that thousands of Syrians and Iraqis have died on the seas trying to reach Europe. The survivors arrive at aid camps tired, hungry, thirsty and scared — and their ordeals are far from over. They are now in the arms of a government without the resources to care for them. "I risked my life because I was very concerned for my children and there was almost nothing inside Fallujah — no food, no electricity, no fuel, nothing," said a woman who arrived recently at a camp in Amiriyat Fallujah, a government-held city south of Fallujah, and gave her name as Umm Bariq. But in the camp, she said, there are shortages of food, medicine and clean water. "So we are suffering here under difficult conditions," the woman said. "We need help here." As bad as conditions are for civilians caught up in the battle, they are likely to get much worse. At least 50,000 civilians are still trapped inside Fallujah under Daesh rule — perhaps 20,000 of them children, according to the United Nations. Last week, the worry was that they would be killed in the crossfire as Iraqi forces and their Shiite militia allies stormed the city. Now, as the fighting has stalled on the outskirts in the face of fierce resistance by Daesh, a siege lasting weeks or months, in the heat of summer, could lead to mass starvation. There have been frantic negotiations, through intermediaries, between international aid agencies and Daesh officials inside Fallujah, seeking to open up corridors to deliver food and medicine. The problem is made worse, some aid workers say, by the Iraqi government's tight control of traffic between Baghdad and Anbar Province, which has delayed the delivery of humanitarian supplies to the camps and the Fallujah area. Out of security concerns, the Baghdad government, which is Shiite-led, has long restricted the movement of people between the capital and Sunni-dominated Anbar, almost as if the two areas were separate countries. For some agencies, it can be difficult and time-consuming to receive permission from the Iraqi government to travel across the bridge and deliver aid. In some instances, aid agencies have turned to a powerful Shiite militia, Kataib Hezbollah, which is controlled by Iran and in charge of an important checkpoint in Anbar, to get aid to the displaced. Also, as the offensive for Fallujah unfolded, the Iraqi authorities were eager to facilitate access for journalists to the front lines, but have not allowed them to travel to areas to see displaced civilians. It was only because of an invitation to join a local aid agency's convoy on Sunday that a reporting team from The New York Times was able to visit the camps for civilians fleeing the violence around Fallujah. The government-controlled areas of western Anbar Province, a Sunni-dominated region that has been a heartland for Daesh, have become vast wastelands of human suffering. Bare-bones tent cities are sprouting up all over, providing little more than basic shelter and some, but not nearly enough, food, water and medicine. The heat is terrible, always well above 100 Fahrenheit during the day, and most tents do not have fans or the electricity to run them. When Iraqi forces reached his town of Saqlawiya, north of Fallujah, last week, Hatem Shukur waved a white flag to catch their attention. In an interview, he said he and his family had been given cold water, watermelon, apples and bananas — delights after months of being under siege. "But now we are facing another problem," Shukur, 58, said. "Can you imagine your family living here in this heat?" He waved his arm around the space where he and his family live, a small square of concrete floor, a metal frame and plastic sheeting for walls. On the floor, lying on a blanket, was his 8-month-old granddaughter, Rawan, flies buzzing around her as she slept. "It just doesn't make any sense to have invested so much in a military campaign to defeat Daesh and not provide lifesaving support to Iraqis in their hours of greatest need," said Lise Grande, the United Nations' top humanitarian official in Iraq. Ms. Grande, who emphasized that the United Nations has not had difficulty with the Iraqi government in arranging deliveries of aid to Anbar, said the effort still faced a steep shortfall in funding from international donors. At the beginning of this year, the United Nations said it needed at least $860 million for pay for urgent humanitarian programs in Iraq. But so far the agency has raised just 30 percent of that sum, roughly $260 million, and is preparing to close down some vital programs this summer. Shiite militias have played a prominent role in the offensive to retake Fallujah after nearly three years of Daesh rule. But because of that, the battle is playing out amid persistent worries that the campaign could intensify the sectarian tensions that are tearing the country apart. Daesh extremists have warned civilians that the Shiite militias would slaughter them in revenge attacks whenever possible. But for the most part, civilians who have fled the areas around Fallujah have said they had tired of the grim life under Daesh and had been treated well by the militias and Iraqi soldiers. "We were surprised that they treated us so well," said a man at a camp who was in his 50s and gave his name as Abu Muhammad, standing on Sunday outside his tent. "Daesh had told us the Shiites wanted revenge and would kill us." Instead, he said, he was given cookies and orange juice. Many civilians have lost their lives trying to escape, either shot by the Daesh, drowned in the river or felled by thirst and hunger during the hazardous hike to safety. Dr. Hassan Abdulfatah, the director of the Amiriyat Fallujah hospital, said he had received 13 bodies of drowning victims over the weekend, including children, and of four others who he said had died of starvation or other ailments trying to reach safety. In every tent scattered across the many camps in Anbar is a sad story, but some are truly wrenching. In one there was a group of grieving women whose children had drowned the night before. One said she had lost three children: Suad, Suzan and Yacoob. Another woman said she had managed to make the river crossing, only to see her daughter and son-in-law drown as they tried. "I was watching them," she said. "I was yelling." — The New York Times