The most senior general in the US armed forces has made an unannounced visit to Baghdad to meet American troops helping Iraqi and Kurdish soldiers in their fight against Islamic State militants.

The visit by Army General Marin Dempsey, chair of the US Joint Chiefs of Staff, is the first by a US general to Iraq since a US-led coalition began air strikes against Isis targets there.

He will meet US officials during his visit, including Kuwait-based task force commander Lieutenant General James Terry, as well as Iraqi officials.

“I want to get a sense from our side about how our contribution is going,” Dempsey said shortly before landing in the capital. “I want to hear from those actually doing the lifting that they’ve the resources they need and the proper guidance to use those resources. This will work best if we’re enabling [Iraq’s] plan.”

Last week, Barack Obama authorised the dispatch of another 1,500 troops to Iraq, nearly doubling the total as the US expands its advisory mission.

Earlier this week Dempsey told the House of Representatives armed services committee that the US was considering the direct use of troops in some upcoming battles against Isis.

Although Iraqi forces were doing a better job now, he said an attempt to move into Mosul or to restore the border with Syria would require more complex operations.

The US military presence in Iraq would remain modest, he told the committee: “I just don’t foresee a circumstance when it would be in our interest to take this fight on ourselves with a large military contingent.”

Iraqi troops have made a series of advances this month, including recapturing areas around the country’s biggest refinery, near Baiji.

That has been the latest setback for Isis, which has had hundreds of fighters killed by air strikes in Iraq and Syria, particularly around the Syrian town of Kobani. Kurdish fighters made significant progress in their defence of the town on Saturday, it was reported.

However, Isis has claimed responsibility for suicide bombings, including attacks in Baghdad, and a car bomb which killed five Iraqi soldiers on a road just north of the capital.

An audio message purportedly recorded this week by the group’s leader, Abu Bakr al-Baghdadi, urged supporters in Saudi Arabia to take the fight to the rulers of the kingdom, which has joined the US-led coalition in mounting air strikes against the militants in Syria.

The strikes have helped galvanise support for Isis among some Muslims in the west, prompting thousands to travel to Iraq and Syria to join the jihad.

The British prime minister, David Cameron, said on Friday that Isis fighters would be prevented from returning to Britain for two years and only allowed to re-enter if they consent to face trial, home detention, regular police monitoring or go on a deradicalisation course.

Meanwhile, on Saturday two parked car bombs exploded minutes apart in an area north of Baghdad, targeting a checkpoint staffed by army soldiers and security forces, authorities said. The blast killed six people and wounded 25, police and hospital officials said.