Thousands of families have been forced to flee the worsening conflict in eastern Ukraine, where the situation in Luhansk and other besieged areas is now "critical", said aid agencies and local officials.

The Red Cross (ICRC) said living conditions for those trapped by fighting between Ukrainian forces and pro-Russian rebels has worsened. Ukrainian troops have made rapid advances in recent weeks, shrinking the territory controlled by separatists. They have also severed the land connection between Donetsk and Luhansk, the last two major cities in the east under rebel control.

But the offensive has caught civilians as well. Luhansk, a city about 20 miles from the Russian border with a pre-war population of 450,000 people, has shrunk to 250,000 since early April and the start of Ukraine's "anti-terrorist operation". It has suffered from repeated shelling. On Tuesday the city authorities said Luhansk had been without water, light, electricity and phone connections for 10 days.

Luhansk was now in a state of "total blockade", they said, with residents trapped in their homes as armed clashes raged around them. Most shops were closed, though bakeries remained open, and municipal rubbish collection continued, officials said. Pensions and other social payments had ceased.

Hundreds of residents have escaped by fleeing into Russia or to surrounding areas controlled by Ukrainian government forces, outside the conflict zone. In the village of Schastya, 14 miles north of Luhansk, locals told Organisation for Security and Co-operation in Europe (OSCE) monitors the city was being shelled "practically non-stop" from 4am to 2am.

They said pharmacies were closed, with people running out of supplies. Drinking water and bread were almost impossible to buy. People were forced to bury bodies in gardens, since funeral services no longer operated. Public transport wasn't functioning either, with only a few ambulance teams still working. Only those looking after bedridden relatives, or those without money, wished to stay in the city, OSCE reported.

The situation in Donetsk, previously a city of 1 million, was less dire, locals said, though it too was under fire. Last week, OSCE observers reported damage to two high-rise residential buildings and a hospital in the city centre, and reported meeting "traumatised and crying civilians and medical staff".

They saw the corpse of a middle-aged man in the city's morgue, with trauma injuries consistent with mortar fire. Some people had taken shelter in the basement of Donetsk railway station. Trains out of Donetsk were still running, but with services sold out for several days. Power and lights were on. A curfew was in force.

The OSCE also said that Ukrainian troops stationed next to the Russian border were coming under frequent rocket attacks. In Krasnyi Derkul, a village 34 miles north-east of Luhansk, observers saw smouldering forest and shrapnel fragments around the area where a border guard unit had built underground shelters. They also saw four shelters in an open field at the edge of the military camp.

"We know the humanitarian situation in eastern Ukraine is rather bad. There has been ongoing fighting for the past few weeks and we know cities such as Luhansk have been cut off from supplies for the past month. The situation is dire in terms of medical supplies, access to water, to electricity, everything," Laurent Corbaz, the ICRC's head of operations for Europe and Central Asia, said.