Exasperated with ragged government relief efforts, Turkish businesses, volunteer organizations, trade associations and individuals have stepped in to provide massive assistance for victims of the crippling earthquake whose death toll rose beyond 10,000 today and was still climbing.

The extraordinary outpouring of local and foreign volunteers, plus cash and material contributions, generated the first sense that, although Tuesday's disaster is far from over, Turkey might be beginning to come to grips with it. "At long last," exclaimed the headline in the leading newspaper Hurriyet, expressing relief after four days of organizational mayhem and despair.

Even as food, water, medicines and supplies were rushed to the survivors, however, Turkey was staggering from its losses, burying its dead and bracing for more bad news. In Geneva, an official at the United Nations raised the specter of a higher death count in the days to come, repeating that the government in Ankara, Turkey's capital, has said that as many as 35,000 people may still be buried in the rubble.

In the hardest-hit areas, located east of Istanbul and closer to the quake's epicenter in Izmit, health officials warned of the possibility of epidemics. They said crushed sewage lines, shortages of running water and portable toilets, and piles of garbage in the streets intensified the risk of cholera, dysentery and other infectious diseases. Medical teams immunized rescue workers against typhoid, sanitation teams fanned out to collect the trash and water purification tablets were distributed.

Still, death was literally in the air: In the most devastated neighborhoods of ruined cities like Sakarya, the smell of rotting flesh hung heavily in the blazing summer heat. There, the authorities buried 963 corpses in a mass grave, photographing each of them so they could be identified by their families.

"We can't cope with this," Oguz Titiz, a doctor, told Turkey's NTV station from Sakarya, 90 miles southeast of Istanbul. "Vomiting and diarrhea started appearing last night, especially among children and the elderly."

Said Turkish Prime Minister Bulent Ecevit: "The greatest problem now facing us is that of epidemic."

In cities, towns and villages in a 100-mile-long destruction zone, millions of people continued to live and sleep outdoors, terrified by an expert's warning that another quake could visit even more destruction on the area. In their tents and crude lean-tos, they populated traffic circles, parks, town squares and beaches. In the city of Bursa, 60 miles south of here, the main stadium, school grounds and other public buildings were turned into vast dormitories.

Aftershocks continued to be felt in western Turkey and the governor of Istanbul was quoted as saying he planned to sleep in his garden. "Nightmare Night," the tabloid newspaper Star called it. "Turkey pours itself onto the streets."

Turkish officials, whose initial estimates of the numbers of deaths were low, generally refrained from guessing at the final toll. An official in Ankara's crisis center, quoted by Reuters news service, said: "The figures are horrendous. Very many people are under the rubble. Many more than anticipated."

Ecevit, who appeared before journalists wearing black, said only that the 45-second quake, which measured 7.4 on the Richter scale, was "one of the worst in human history and the heaviest in Turkish history."

Estimates of the costs to Turkey's economy ranged from $5 billion to $40 billion.

If there was a glimmer of hope, it was that an emergency relief effort was beginning to work on something approaching a scale commensurate with the disaster that has befallen this country of 63 million people. Much of it had nothing to do with the government:

* In Istanbul, the luxurious Ciragan Palace Hotel sent hundreds of hot meals into suburban neighborhoods that were badly damaged in the quake.

* The Turkish Soccer Federation announced it had donated about $1.2 million to victims of the temblor, and one of the country's most popular teams, Besiktas, said it will play a benefit match against Germany's Bayern Munich to raise more cash.

* Several insurance companies said they will cover treatment for quake-related injuries even if exempted by individual policies.

* A volunteer who staffed the crisis center's hot line in Istanbul said many phone calls he fielded were not from victims but rather from volunteers eager to help them.

* Universities, dairies, soft drink companies, industrial holding companies and banks kicked in with donations as small as a minibus full of medicine and as large as a donation of $2.5 million in cash.

Many volunteers said they were driven not only by altruism but also by a perception that Turkey's government is simply not up to the job.

"It's very simple," said Huseyin Beyazit, a 39-year-old telecommunications executive who organized donations of money and supplies from an association of young businessmen in Avcilar, 20 miles west of Istanbul. "The government doesn't have a national emergency plan. There's no preparedness."

Said his brother Husamettin, 31: "We're all involved, trying to help, but there's no coordination from the government. It's like a big ocean and everyone's throwing in a rock."

In addition to local aid and volunteers, assistance has poured into Turkey from Japan, Kuwait, Saudi Arabia, Belgium, England, the United States, Greece, Israel, Azerbaijan, Germany, Italy and the United Arab Emirates. The International Monetary Fund gave Turkey a $325 million loan.

Popular anger directed at the government tended to be most intense in the most devastated communities, where grief and emotions ran high. In neighborhoods spared the brunt of the quake's force, the government is regarded more sympathetically--as overwhelmed, but not necessarily inept.

Still, the quake and the government's faltering response have exacerbated a traditional geographic and social divide in Turkey between the heavily industrial and often industrious west, centered around Istanbul, and the somewhat less prosperous east, with its focal point in the political capital of Ankara. In this schism, the resentment runs heavily from west to east; Turkey's productive heartland often takes a dim view of the capital, its political machinations and frequent changes of government.

"In the first couple of days [after the quake], we looked by habit for the state to rush to our aid," wrote Enis Berberoglu in Hurriyet. "But it was futile, because the state can't afford to save itself, let alone us. But [the private sector] is different--much better off and stronger than the state."

Turkey's governments in recent years have lasted on average only a couple of years. Despite the criticism, the current coalition government appears so far to be in no danger of being toppled in fallout from the quake. The opposition has called the relief effort incompetent, but, perhaps because it would be unseemly to make political hay of such suffering for now, the criticism has been muted.

"A lot of the criticism of the government is emotional--'How could this happen? Somebody has to be blamed'--and there have been obvious glitches and organizational problems," said a diplomat in Ankara. "But barring Murphy's Law going into overtime, I think they'll get through this."

CAPTION: Two women mourn at the grave of a relative killed in the earthquake, whose death toll passed 10,000 yesterday. The Turkish government said as many as 35,000 more people may be buried in rubble, and the widespread destruction of sanitary services could cause epidemics of cholera, dysentery and other diseases.