Despite military triumphs, Islamist militants are losing hearts, minds and obedience of residents who have had enough

Iraqis living under Isis rule in Iraq, where non-Sunni residents have been forced from their homes and tens of mosques have been deemed idolatrous and marked for destruction, have started to push back against the extreme interpretation of Islam being imposed on them.

The Islamic State in Iraq and the Levant has won significant territorial victories and declared an Islamic caliphate in swaths of land it has seized, from al-Bab in Syria to Falluja in Iraq. The US recently said Isis was worse than al-Qaida (pdf) and that it had a "full-blown army". It has subsequently increased reconnaissance flights over Mosul, from one flight a month just two months ago to 50 flights a day.

Isis fighters have fought and wrested territory from the Syrian army, the Iraqi army and the Kurdish peshmerga, but have revealed their fragility in governance, in particular, a brutal disregard for local religious and cultural values.

In Mosul, despite its military triumphs, Isis is losing the hearts, minds and obedience of residents who say they have had enough.

When its fighters destroyed the Nabi Jonah mosque (Jonah's tomb) in the Iraqi city last Thursday, they failed to removed copies of the Qur'an and other religious texts. Residents treading through the ruins of the building found torn and burnt pages of the holy books scattered across the rubble. It was an insult to Islam that was captured on video and unified the city in outrage.

"[Isis] claims that having graves inside mosques is heretical but what about the Qur'an, why did not they remove the Qur'an from the mosque before destroying it?" one resident, who did not wish to be named, asked the Guardian.

The fighters – who adhere to an extreme interpretation of Sunni Islam that requires the destruction of shrines and graves as idolatory – have reportedly drawn up a list of around 50 mosques to be destroyed in Mosul so far.

The group has a unit called Katayib Taswiya, the demolition battalion, whose job is to identify heretical mosques for destruction. The battalion razes to the ground any mosques built on tombs. If a graveyard has been built after the mosque's construction, then they will destroy the graves and any section of the mosque building.

Among the 50 on the list are a shrine to the prophet Seth – considered in Islam, Judaism and Christianity to be Adam and Eve's third son – and the 14th-century Prophet Jirjis mosque and shrine, which was bombed and largely destroyed on Friday. The Prominent Iraqi architect Ihsan Fethi described the destruction of the heritage site in Nineveh as "cultural suicide".

Speaking to the Guardian from Mosul, Bashar, a 38-year-old musician, said people had tried to occupy the mosques under threaten in an effort to prevent fighters from bombing them.

When the demolition battalion made its move on the Jirjis mosque in the Souq al-Sharin neigbourhood, some residents decided to take a stand. On Friday and Saturday evening, they slept inside the mosque in the hope that their presence would dissuade the militants from their demolition attempts. The fighters came back on Sunday and destroyed the graveyard as planned, but most of the mosque is still standing.

Isis defended its destruction of the sites in a post on one of its main websites on Tuesday: "The demolition of structures erected above graves is a matter of great religious clarity. Our pious predecessors have done so … There is no debate on the legitimacy of demolishing or removing those graves and shrines."

But on Sunday, Mosul residents continued their defiance. They had named Monday as the first day of Eid al-Fitr, marking the end of the fasting month of Ramadan. On Sunday evening, militants paraded through the city, ordering citizens through loudspeakers mounted on vehicles to continue their fast on Monday or face punishment. These warnings were ignored and the arrival of Eid was announced from Mosul's mosques on Monday. In the face of public rebellion, Isis changed its mind and several hours later announced the end of Ramadan.

With at least 8,000 years of continuous habitation, Mosul is considered an archeological treasure, with many heritage sites belonging to all religions and sects. Dubbed "small Iraq", people from a range of religions and ethnicities have lived side by side peacefully for centuries.

This solidarity was displayed last week when several thousand Christian residents were given a deadline of midday on Saturday to convert to Islam, pay a special tax or "face the sword". Fleeing Christians told the Guardian that when they were preparing to leave, fearful of the threats, their Muslim neighbours told them to stay put and promised to defend them should Isis come after them. Most of the Christian population fled regardless to areas under control of the Kurdistan regional government.

This weekend, reports leaked from the city that Isis had ordered the closure of women's salons and placed specific restrictions on the styling of men's facial hair. Drug supplies, particularly for those with kidney disease, are running short.

In what could be an indicative violent eruption of resentment and anger from the population, two Isis fighters were reportedly shot dead in broad daylight in the Qayara neighbourhood of south Mosul on Sunday. A witness told the Guardian he saw three assailants fleeing the scene through the city's narrow alleyways.

The initial joy with which Isis was received in Mosul, as liberators for the Sunni population after years of sectarian corruption and restriction at the hands of the Iraqi army, may already have run dry.