Fighting between Ukrainian government troops and pro-Russian militia is fuelling a worsening humanitarian crisis in eastern Ukraine. Tens of thousands of people are fleeing combat, most of them from the rebel capital of Slavyansk, where almost daily shelling has claimed numerous civilian casualties since late May.

Most residents in the besieged city have been without water, electricity and gas for the past week. Food supplies are limited, and grocery stores smell of rotting food from the lack of refrigeration. Dozens of people queue for drinking water.

About 270 people have died in the east of the country since Kiev launched its "anti-terrorist operation" two months ago, Ukraine's health ministry said on Wednesday. Of those killed, 225 were in Donetsk region, including Slavyansk, where fighting has been heaviest. At least two children have died of shrapnel wounds in Slavyansk this month, according to the health ministry. However, past government estimates have been low, excluding deaths in rebel-held territory.

Appalling conditions in rebel-held towns have caused thousands to flee. The exodus from Slavyansk gathered pace when Ukrainian army shelling intensified at the end of May, with most residents going to the nearby city of Svyatogorsk were they are dependent on the goodwill of locals for housing and food. About 15,000 to 20,000 refugees from Slavyansk have arrived in the city since the end of May, according to mayor Alexander Dzyuba.

Svyatogorsk monastery – one of the holiest Russian Orthodox sites in eastern Ukraine – has been housing as many as 500 Slavyansk inhabitants each day, said one monk. The displaced people receive one meal a day and live in tight quarters. Women and children live three to a room in the female section, while men live seven to a room with only a few feet between each bed. Laundry and clothing hung from lines above the cots.

Several old recreational compounds in the area have also been taking in displaced people. Volunteers hand out limited portions of donated food, much of it from residents in the eastern city of Kharkiv and from the company Bravo, at the Cafe Pyramid, said volunteer Yelena Laskova.

Sergei and Yulia Rivokovsky were staying with their three young children at a "tourism base" called Iskra. Yulia and her daughters arrived by bus two weeks ago, while Sergei rode his bicycle nearly 40 miles to join them. He said he wanted to join the rebels but had to see his family first. "My daughter was crying over the phone. If it gets worse I will go fight."

"There's no prospects for the future here," said Slava, who was staying with four relatives in a room at Iskra. "There's no work. Where can we go? But at least there's a roof and lighting here."

Another nearby recreational compound called Dubravushka was housing 170 displaced people from Slavyansk, according to manager Svetlana Sachenko. One of them, Anna Ergert, said her son and elderly parents remained in Slavyansk because they hadn't wanted to leave.

"My dad doesn't want to eat or drink, he wants to die … I tried to bring him borscht, but he said, 'You don't bring borscht to a corpse,'" she said, breaking down in tears.

According to the health ministry, 251 hotels, summer camps and other sites across the country have been converted into refugee centres housing up to 30,000 displaced people – including the servants' quarters at deposed president Viktor Yanukovych's luxury estate in Kiev. More than 7,000 people have already moved into these temporary housing facilities.

But no government assistance was visible in Slavyansk or Svyatogorsk, where local civic organisations, businesses and private citizens have helped to provide transportation, housing and food for those trying to flee the violence.

According to local volunteer Natasha Bogamaz, the Communist party has been organising buses to take women and children out of Slavyansk since 3 June. Buses were to take 20 people to the nearby city of Chuguyev and 40 to Crimea on Wednesday.

Anna Ivashenko was waiting to take the bus to Crimea with her two-year-old and four-year-old daughters. They decided to flee after a shell exploded in their apartment yard the day before, injuring her father. The family lives outside the village of Andreyevka, where there has been especially intense fighting, and her "nervous system was shot", she said.

Newly elected president Petro Poroshenko has promised a humanitarian corridor in the east, though there has been no sign of this in practice yet. On Thursday the security service ordered the commander of Kiev's "anti-terrorist operation" to establish a corridor for the evacuation of civilians.

Russian officials have also claimed that about 70,000 Ukrainian refugees have fled across the border since the fighting began, though these figures appear to have been inflated. The secretary general of the OSCE, Lamberto Zannier, visited refugees in Rostov region on Thursday, and recommended that the United Nations High Commissioner for Refugees get involved in helping those people who have fled.

Although thousands have fled, a majority of Slavyansk's 120,000 population has remained, unwilling or unable to leave because of a lack of funds, lack of transportation or fear of the troops.

In central Slavyansk on Wednesday, pensioner Valentina Vasiliyevna was clearing glass and shattered furniture out of her flat, which was hit by a shell earlier this week just after she and her son went to bed. "Thank God our beds are behind a wall, and we weren't killed," she said. "I lived through the great war, and now we're living through another one."

It was not clear whether the shell was fired by Ukrainian or rebel forces, although government troops have fired more ordinance. The commander of a government checkpoint just outside Slavyansk who would identify himself only as Mikhail said rebels had been firing long-range weapons from residential buildings, while Yura, a rebel, said the military was "shooting at peaceful people".

Howitzer shells, mortars and rockets have struck numerous buildings in Slavyansk and nearby towns. A block of flats on Lenin Street was rendered uninhabitable after being hit at least three times, and residents said civilians had been killed in the blasts. Anatoly Zhuly said he was going to live with his daughter inn the city's outskirts. Asked how he could stay in a city being shelled every day, he said: "If you're destined to be hanged, you won't drown."

Leonid Kozmenko, one of the few still living in the wrecked building, said he couldn't leave. "We would need money first," he said. "We have nothing."