GAZIANTEP, Turkey — The Islamic State’s vaunted exercise in state-building appears to be crumbling as living conditions deteriorate across the territories under its control, exposing the shortcomings of a group that devotes most of its energies to fighting battles and enforcing strict rules.

Services are collapsing, prices are soaring and medicines are scarce in towns and cities across the ‘‘caliphate’’ proclaimed in Iraq and Syria by the Islamic State, residents say, belying the group’s boasts that it is delivering a model form of governance for Muslims.

Slick Islamic State videos depicting functioning governing offices and the distribution of aid fail to match the reality of growing deprivation and disorganized, erratic leadership, the residents say. A trumpeted Islamic State currency has not materialized, nor have the passports the group promised. Schools barely function, doctors are few and disease is on the rise.

In the Iraqi city of Mosul, the water has become undrinkable because supplies of chlorine have dried up, said a journalist living there, who spoke on the condition of anonymity to protect his safety. Hepatitis is spreading and flour is becoming increasingly scarce, he said. ‘‘Life in the city is nearly dead, and it is as though we are living in a giant prison,’’ he said.

In the Syrian city of Raqqa, the group’s self-styled capital, water and electricity are available for no more than three or four hours a day, garbage piles up uncollected and the city’s poor scavenge for scraps on streets crowded with sellers hawking anything they can find to sell, residents say.

Videos filmed in secret by an activist group show desperate women and children clamoring for handouts of food, while photographs posted on the Internet portray foreign jihadists eating lavish spreads, a disparity that is starting to stir resentment.

Much of the assistance that is being provided comes from Western aid agencies, who discreetly continue to help areas of Syria under Islamic State control. The United States funds health-care clinics and provides blankets, plastic sheeting and other items to enable the neediest citizens to weather the winter, a U.S. official said, speaking on the condition of anonymity because of the sensitivity of the subject.

The government workers who help sustain what is left of the crumbling infrastructure, in Syrian as well as Iraqi cities, continue to be paid by the Syrian government, traveling each month to collect their pay from offices in government-controlled areas.

‘‘ISIS doesn’t know how to do this stuff,’’ said the U.S. official, using an acronym for the group. ‘‘When stuff breaks down they get desperate. It doesn’t have a whole lot of engineers and staff to run the cities, so things are breaking down.’’

There are also signs of falling morale among at least some of the fighters whose expectations of quick and easy victories have been squashed by U.S.-led airstrikes. A notice distributed in Raqqa this month called on fighters who were shirking their duties to report to the front lines, and a new police force was created to go house-to-house to root them out.

There is no indication that the hardships are likely to lead to rebellion, at least not soon. Fear of draconian punishments and the absence of alternatives deter citizens from complaining too loudly, the residents said, in interviews conducted while they were on visits to neighboring Turkey or over the Internet.

But the deterioration is undermining at least one important aspect of the Islamic State’s self-proclaimed identity — as a state, dedicated to reviving the 7th-century caliphate that once ruled the Muslim world. Governing is as central to that goal as the military conquests that occurred as Islamic State fighters swept through much of Syria and Iraq over the past year.

The group’s momentum on the battlefield has been slowed by the U.S.-led air campaign, which has helped reverse or stall Islamic State offensives on numerous fronts, from the tiny town of Kobane in northern Syria to the farmland south of Baghdad.

That the group is also failing to deliver services in the areas it does control calls into question the sustainability of its larger ambition.

The Islamic State ‘‘is not this invincible monster that can control everything and defeat everyone,’’ said an activist in the eastern Syrian city of Deir al-Zour, who also spoke on the condition of anonymity to describe the ineffectual delivery of services there.

‘‘The whole idea that it is well organized and an administrative entity is wrong. It is just an image.’’

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It is in Raqqa, the first major city to fall under Islamic State control more than a year ago and the cradle of its governance experiment, that the discrepancy is perhaps most conspicuous. A Raqqa businessman who traveled to Mosul recently said the Iraqi city is in far better shape than his own city in Syria, where people are being driven away by the specter of hunger and devastating government bombing raids that have killed mostly civilians.

The bombardments have played a big role in straining the infrastructure. U.S. airstrikes, aimed at Islamic State targets, have also contributed, forcing the group to abandon many of its government buildings. American attacks on the small, makeshift oil refineries that many citizens relied on for income have deepened the deprivation, leaving many people without income and sending prices soaring.

Whether the Islamic State’s administration was ever as capable as it has been portrayed appears to be in doubt, Syrians say. Those who could afford to flee areas controlled by the group have done so, disproportionately including the professionals and technocrats whose skills are needed to run government services.

Syrians say the Islamic State’s administration is overseen by a network of shadowy emirs or princes. Lower-level positions are occupied by Syrians or foreigners who often lack administrative or technical skills.

‘‘ISIS has become too big to control itself,’’ said a Syrian aid worker who regularly interacts with Islamic State officials and who did not want to be identified in order not to compromise his dealings with the group. He finds them willing and cooperative, ‘‘but they’re not smart, and they’re not capable. They have no expertise.’’

For most citizens, the main interaction with the Islamic State is with its ubiquitous police and security agencies, including the notorious Hesbah, which patrols the streets in quest of those transgressing the group’s harsh interpretation of Islamic law.

Those rules continue to be rigidly enforced. Shopkeepers shut their stores five times a day for prayer. Smokers have quit for fear of the obligatory three-day jail sentence for a first offense — and a month for a second. Public executions for theft, blasphemy and dissent are on the rise. A new punishment, for homosexuality, in which the accused is thrown off a tall building, has been implemented twice in recent weeks.

Meanwhile, crime has plunged, and for many residents the order is a welcome alternative to the lawlessness that prevailed when more moderate Syrian rebels were in charge. Syrians who lived for decades under the regime of President Bashar Assad are accustomed to obeying orders, and many have adapted to the new rules, said a government employee in the former tax department who collects his salary from the government, even though he is no longer working.

‘‘Daesh are not as cruel as the regime was,’’ he said, using an Arabic name for the jihadists. With the Islamic State in charge, ‘‘if you don’t do anything wrong — according to their standards, not ours — they will not bother you.’’

The strict enforcement of rules sometimes undermines efforts to deliver services, however. When electricity workers raced to repair cables damaged by government shelling in the town of Deir al-Zour, the Islamic State detained and lashed them for violating the prohibition on working during prayer time, said the Deir al-Zour activist.

The entire staff of one of the city’s four functioning field hospitals was detained as they held a meeting because three of them were smoking.

There is no indication that the Islamic State’s income, estimated at $12 million a month, is suffering. Syrians continue to sign up because there are no other jobs available, residents say.

Islamic State functionaries also continue to exact payments, going door to door to collect taxes from shopkeepers and fees for electricity and telephones.

‘‘If the regime did not supply telecoms and salaries, I don’t think ISIS could survive,’’ said Hassan Hassan, a Syrian analyst with the Abu Dhabi-based Delma Institute. ‘‘It charges people for things the regime is providing. But it’s not viable as a state.’’

Tensions are emerging between the local populace and the foreign fighters, estimated by U.S. officials and analysts to number around 15,000, or about half of the total fighting force. Foreigners get paid in dollars, while Syrian recruits, known as munasir, or helpers, are paid in Syrian pounds.

Islamic State fighters get treated in their own secretly located field hospitals, while civilians are forced to rely on the collapsing private hospitals, said Abu Mohammed, an activist with Raqqa is Being Slaughtered Silently, a group that works to draw attention to conditions under the Islamic State. He uses a nickname to protect his safety.

‘‘People are fed up with them and would like to get rid of them,’’ he said. ‘‘But they don’t have the ability.’’