The beleaguered inhabitants of Raqqa, self-proclaimed capital of the Islamic State (Isis), are suffering widespread hunger, crippling inflation, chronic power shortages and poverty so acute that emergency soup kitchens have been set up.

With no journalists, local or foreign, able to operate inside Syria’s sixth-largest city, courageous local activists have given the Observer a detailed account of life under the jihadists’ totalitarian regime, a rare glimpse of everyday life in the city.

Their testimony reveals the evolution of a community brutally divided into haves and have-nots, with Isis enjoying well-resourced services including “private” hospitals and a relatively high standard of living as many residents struggle to make ends meet.

Crucifixions of Isis opponents have taken place in Raqqa’s Paradise Square, as well as frequent beheadings and lashings for offences as minor as smoking a cigarette. Abu Ibrahim Raqqawi, founder of a network of activists called Raqqa Is Being Slaughtered Silently, told the Observer: “Isis kills a lot of people, we see a lot of executions, a lot of beheadings. I have seen about five people crucified in the city. People are now calling Paradise Square Hell Square.”

Children collect items from among the debris of a school for the deaf and mute, destroyed in what activists said were overnight US-led air strikes in Raqqa. Photograph: NOUR FOURAT/REUTERS

Meanwhile the population is subjected to a rolling daily nightmare as the Syrian government launches air strikes in the morning, which are followed by coalition air raids in the evening.

When warplanes belonging to President Bashar al-Assad killed scores of residents last week, repeatedly targeting heavily populated areas of the city, Isis and its battery of anti-aircraft and missile defences did not fire a single round. The lack of response further enraged Raqqa’s inhabitants towards the extremists, according to witnesses.

“People are getting very angry because Isis do not shoot at the aircraft with their rockets, they just watch the people die. We have a situation where there are Syrian air strikes at the start of the day and coalition air strikes later and in between Isis is controlling and killing the people. Everybody is tired and afraid,” said Raqqawi.

The reports of the activists running Raqqa Is Being Slaughtered Silently offer an insight into how the jihadists intend to run their embryonic caliphate, which so far has around six million people under its rule in northern Syria and northern Iraq.

Raqqawi portrays a dysfunctional economy presided over by an untouchable elite with extremists able to live handsomely as thousands of civilians struggle to afford basic food. The price of bread has risen 150%, from 37p to 94p or 250 Syrian pounds since September. Meanwhile, Isis fighters boast about drinking Red Bull, also costing 250 Syrian pounds a can, and getting paid a stipend equivalent to more than 30,000 pounds of local currency a month, around twice as much as the average wage of Syrians in that part of the country.

Despite the growing disparity in living standards between Raqqa’s residents and Isis – which reportedly earns more than $3m (£2m) a day in black market oil sales – the extremists do not appear interested in distributing their wealth to win favour with the local population.

“We have something here we call kitchen relief, like a soup kitchen, which gives one free meal a day to the people and has more than 1,000 families using it. Isis do not give this kitchen anything,” said Raqqawi.

Children at a jihadist training camp in an image released by activist network Raqqa is being Slaughtered Silently. Photograph: AP

Water has also become a precious commodity, with some families forced to obtain it from the Euphrates river, according to Raqqawi, after coalition air strikes destroyed the oil refineries and power supply to the city’s water pumps. Raqqawi said that the US air strikes that began hitting the city almost two months ago have ended up adversely affecting the civilian population and not just their intended target of Isis.

Speaking from Raqqa on Friday, Raqqawi said: “The city is suffering from poverty and disease. A big problem is that all the prices inside the city have become very expensive especially after the coalition air strikes. There is no electricity, everyone is dependent totally on the generators.

“When coalition air strikes destroyed the oil refineries inside the city, prices grew threefold. The money that the people have is not enough to buy food, which has become very expensive.”

By contrast Isis, he said, are living in relative luxury. He said that their generators are permanently turned on, whereas in the rest of the city, power is restricted to between three and five hours a day.

In addition, Isis-only hospitals are staffed with the best doctors and the latest equipment while civilians frequently die due to inadequate care: “People are dying from injuries because there is no medical equipment, no supplies, no doctors, no ambulance crews. Isis have their own hospitals where they do not allow the civilians to go. These hospitals have the best medical treatment, the best doctors.”

A child at an Islamic State group training camp in Raqqa. Photograph: AP

Activists, located in safe houses dotted throughout the city, use encrypted conversations online to exchange intelligence and evade attempts by teams of hackers employed by Isis.

Among their latest plea to the international community are calls for the US-led coalition to thwart air strikes by the Syrian government by imposing a no-fly zone above the city.

On Tuesday air strikes from Syrian government warplanes targeted at least nine sites in Raqqa, including a crowded market near its museum, and heavily populated civilian areas.

The UK-based Syrian Observatory for Human Rights said the air strikes had killed at least 95 people, among them three women and four children.

Raqqawi referred to the air strikes on Tuesday as the bloodiest since the start of the 2011 revolution, describing how two buses were caught up in the attack. All the passengers burned to death. The death toll was more than 200, he said, with 170 bodies taken to the city’s last remaining public hospital; another 50 bodies were burned beyond recognition.

No one, he said, knew precisely how many residents had been executed by Isis but he said that deaths from disease were rising, although he was unaware of any deaths from starvation despite the growing issue of poverty.

Raqqa’s population is still above 200,000 despite the air strikes, and Isis is said to have a residual force of between 3,000 to 5,000 in the city, although numbers change depending on levels of fighting elsewhere in the caliphate.