A fighter with the U.S.-backed Syrian Democratic Forces aims through a peephole in Raqqa, Syria, on June 11, 2017. (Youssef Rabie Youssef/EPA)

As U.S.-backed forces press farther into Raqqa, the Islamic State’s stronghold in ­Syria, human rights groups pleaded Tuesday for the safety of thousands of residents still trapped in the city.

The Syrian Democratic Forces, or SDF, have made swift progress since entering the city last week, despite a U.S. military assessment that the battle ultimately would be “long and difficult.”

Residents reported heavy shelling Tuesday as the SDF fought around the edges of Raqqa’s Old City. The area is fortified by 8th-century walls, a reminder that it was once the capital of the Abbasid caliphate.

Satellite images taken May 20 appear to show that the militants had erected relatively few defenses on the road to the Old City. In an interview with Al Jazeera, an SDF fighter said his unit was attempting to clear the area of booby traps and Islamic State snipers.

Human Rights Watch issued a plea to the U.S.-backed forces to do all they can to protect the tens of thousands of civilians believed to be trapped inside the city. 

“The battle for Raqqa is not just about defeating ISIS but also about protecting and assisting the civilians who have suffered under ISIS rule for 3½ years,” said Lama Fakih, deputy Middle East director at Human Rights Watch, using another name for the Islamic State. 

“Coalition members and local forces should demonstrate concretely that the lives and rights of the . . . civilians in Raqqa are a parallel priority in the offensive,” Fakih added.

Monitoring groups say coalition airstrikes have caused an unusually high rate of civilian casualties in recent months. The SDF also has killed civilians as it shelled Islamic State territory.

The Syrian Network for Human Rights monitoring group published photographs Tuesday of a young girl it identified as Naya Abo Haif, saying she had died in shelling the day before alongside her father and brother. In one image, she stood smiling shyly in a Minnie Mouse T-shirt. Another showed the child’s body wrapped in a green shroud, her face half-masked by blood.

The Office of the U.N. High Commissioner for Refugees is calling for help securing land routes so its staff can reach tens of thousands of displaced civilians across the surrounding province. According to the agency’s figures, more than 100,000 people were on the move in the area during May alone, many of them displaced several times over.

Residents still inside Raqqa say food supplies are dwindling, while running water is available for only a few hours a day. One man described snipers across the city’s rooftops, shooting at anyone who tries to flee.

In neighboring Iraq, the Islamic State is close to losing its onetime stronghold of Mosul, after a months-long battle led by U.S.-trained elite units of the Iraqi army.

The pressures on Raqqa and Mosul, cities the Islamic State once called capitals, have pushed senior leaders to the province of Deir al-Zour, a vast stretch of the oil-rich Syrian desert that sits between those areas. U.S.- and Iran-backed forces have been jostling for position ahead of an offensive to capture Deir al-Zour, hoping to further their own regional ambitions in the process.

For Iran, securing a land route across the Syria-Iraq border to its Lebanese proxy, Hezbollah, is a key motivation. The United States, which lists Hezbollah as a terrorist organization because of its attacks on Israel, would oppose such a conduit.

For the United States, control of Deir al-Zour would be a boost to President Trump’s calls to blunt Tehran’s influence in the region. It also would provide U.S.-backed forces with an important bargaining chip to use in the event of a final peace settlement for Syria.

Although the United States has shown willingness to back its force of Arab fighters with defensive airstrikes, there are few signs that the rebels are strong enough militarily to move northeast to Deir al-Zour.

This weekend, an assortment of Iranian-backed militias appeared to all but end U.S. hopes of reaching the Deir al-Zour town of Bukamal when they looped around a Syrian rebel force supported by U.S. Special Operations forces in a move to cut its planned route.

A despondent U.S.-backed rebel fighter, who spoke on the condition of anonymity because he was not authorized to talk to reporters, said Tuesday that the situation was “not looking good.”

Zakaria reported from Istanbul.