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Rebels argue that the humanitarian assistance is in effect helping Mr. Assad survive the war of attrition. "Aid is a weapon," said Omar Baylasani, a rebel commander from Idlib, speaking during a visit to a Turkish border town. "Food supply is the winning card in the hands of the regime."

The biggest obstacle blocking aid from rebel-held areas is the United Nations requirement that its relief agencies follow Mr. Assad's rules — which limit access to opposition territory — as long as the international assembly recognizes his government. The United Nations agencies are the main conduit for international aid, including most of the total of $385 million that Washington has directed to the cause in 2012 and 2013.

That means that while internally displaced Syrians living in government-controlled areas are cared for in United Nations-run camps, with standard shelter and basic utilities, the many who have fled into opposition territory are plagued by shortages of food, fuel, blankets and medicine. At a civilian medical clinic here in the rebel-held countryside north of Aleppo, the 15 doctors kick out their hundreds of patients each day at 4 p.m. because there is no fuel or power to keep the lights on.

The lack of foreign aid "is a catastrophe," said Saed Bakur Abu Yahia, the clinic's director. "We get nothing, " he said, bundled in a winter jacket and rubbing his hands for warmth as he sat in his office.

The United States has done more than any other country to circumvent the United Nations, but its efforts remain unknown to most Syrians. Washington is funneling about $60 million — about $10 million in 2012, and about $50 million in 2013 — through independent nonprofit groups to deliver flour, food baskets, blankets and medicine to the most stable opposition-controlled territory (one group said it reached most of Aleppo Province, but not yet Idlib). The nonprofit groups insist on keeping their work and their American donors a secret to protect staff members still working in Damascus under Mr. Assad.

"Our humanitarian assistance, $385 million to date, is making a difference," said one United States diplomat involved in Syria policy, speaking anonymously because of the sensitivity of international politics around the aid. "But we can't talk about it."

The result is deepening cynicism and anger toward the West among precisely those Syrians the United States and its allies say they are trying to assist and befriend, a disappointment and frustration to those in Washington who hoped to win favor among the opposition.

In interviews, dozens of Syrians living in rebel-held territory in the provinces of Aleppo and Idlib insisted that their towns had received no Western aid and groused about "empty promises." Only a few most directly involved in aid distribution acknowledged recent visits from international nonprofit groups, and those with knowledge of the meetings insisted that the names of the aid groups remain confidential.

Even the Syrians most involved in the Western effort expressed frustration. "We believe we are owed an explanation over where this money is going, but every time we ask, we can't get an answer," said Ghassan Hitto, who runs the aid coordination arm of the Western-backed Syrian National Council. He estimated that as much as 60 percent of the Syrian population lives outside the Assad government's control and thus beyond the reach of most aid. It is an assessment that is impossible to confirm but feasible because of the heavy population of the rebel-controlled north.

United Nations officials acknowledge the problem. But they say they have few alternatives while the United Nations recognizes the government, which is unlikely to change as long as Russia supports Mr. Assad. "The government, whether you like it or not, is still the government," said Jens Laerke, a spokesman for the United Nations Office for the Coordination of Humanitarian Affairs.

Among other obstructions, Mr. Assad has blocked any United Nations agencies from the shortest and safest route into rebel territory, across the opposition-controlled border with Turkey. "We do not have the government's consent," Mr. Laerke said. "So we cannot, as the United Nations, do that."

To get around that ban, at least three United Nations relief convoys have crossed the battle lines to reach parts of the north, Mr. Laerke said, but it is a dangerous trip. Eight United Nations aid workers have been killed during the Syrian conflict, he said.

Mr. Laerke said that as much as 45 percent of the 1.7 million people inside Syria who have received some aid from United Nations food programs live in areas that are at least contested by the rebels. But other aid workers and international observers said those statistics vastly understated the gap.

"There is a big discrepancy," said Dr. Mego Terzian, Syrian coordinator for Doctors Without Borders, which operates two hospitals at undisclosed locations in the opposition-controlled zone. The medical group has pleaded for more aid to cross the Turkish border.

Hisham Mohamed Bakur, 40, director of the aid office in Sawran, argued that the United Nations had betrayed its own principles of neutrality by allowing Mr. Assad to manipulate the aid. "As a human being, I think it is a good idea to deliver aid to people in need wherever they are," he said. "But to the people, not the government."

Speaking on the condition of anonymity, officials from the few Western nonprofit groups working in rebel-held Aleppo say the violence of the war has also limited their reach.

"The reality is that it is hard to get help to the opposition areas," said an official at a group that is receiving $20 million from the United States.

"We are helping feed 410,000 people, but it is just a drop in the bucket," the official said. "There is just not enough, and every day the crisis gets worse." (Washington has so far financed three nonprofits and recently added two others, officials said.)

But aid workers say that other donors, mainly from Arab states hostile to Mr. Assad, funnel their money directly through the military brigades, each of which now operates its own civilian "relief arm" to further its political goals.

"We are seeing an important amount of money from the gulf countries," said Dr. Terzian of Doctors Without Borders. He added, "But the reality is that most of the financial contributions from these countries are going to war efforts and very little to meet humanitarian needs." Even that, he said, was "for propaganda."

At the rebel-controlled border with Turkey at Kilis, the disparities are jarring. Just outside Syria, the Turkish government runs a state-of-the-art refugee camp that resembles a neat village of refabricated cottages, each with heat, electricity and appliances.

Refugee children attend a bright and spacious elementary school with therapists specializing in post-traumatic stress. The residents use laundry machines, take computer classes or attend lectures at a community center. They buy groceries at two competing markets using credit on specially issued cards verified by high-tech fingerprint scanners.

But just inside the same border is an unplanned camp more squalid than any in Jordan, Lebanon or Turkey.

Thousands — there is no reliable count — sleep crammed in leaking tents without heat or electricity. They crowd like cattle in metal chutes, holding plastic buckets for the crude meals they are served usually twice a day, but sometimes once or not at all. The only aid provider is a small team from the Turkish humanitarian group known as IHH, which is easily the most visible international organization in opposition territory.

Children in the camp slosh through muddy puddles in plastic sandals so big they fall off their feet. On a recent visit, dozens of refugees swarmed around a storeroom where aid workers were dispensing desperately needed new shoes — castoffs from Turkish corporate donors. Abdel Rahman Haji, 29, a burly man with a thick beard, emerged holding up a pair of bright red women's shoes with three-inch heels, size 37.

"Who is going to wear high heels in this mud?" he said.

"Are we dogs that they give us such things?" screamed a woman in a traditional Muslim head covering and cloak, clutching another pair of pumps.

As darkness fell moments later, thick clouds of noxious black fumes poured out of stovepipes sticking through the tents and hovered over the camp. Desperate for heating fuel, the refugees said they burned the shoes for firewood, just as they did the castoff sport coats and sundresses they had also been given.