The Taliban have claimed the capture of Kandahar, Afghanistan’s second-largest city and the place where the hardline Islamist group first emerged in the 1990s, as its fighters rapidly extend their control over vast swathes of the country.

The claim came as Afghanistan’s third city of Herat, in the country’s west, fell fully to Taliban on Thursday with insurgents overrunning government positions and flying their flag over the city. With the Taliban in full control, they promised in a message to residents to “bring security” as its fighters fired into the air in celebration and as local officials fled to an army base in the city’s outskirts.

The latest Taliban advances leave the capital, Kabul, isolated from the rest of the country and facing a rapidly escalating threat as provincial capitals have toppled one after another in the past seven days.

The US and other western countries have accelerated planning for the evacuation of their embassies in Kabul and the US embassy website has ordered its citizens to leave Afghanistan immediately.

The advances came amid reports of a power-sharing offer made by the beleaguered Afghan government’s negotiating team to the Taliban office in Doha, an approach the group has reportedly rejected.

The fall of Kandahar on top of Herat would be a catastrophic blow to the Afghan government which has watched its forces crumple as the Taliban have swept through Afghanistan’s cities in a lightning offensive.

Confirming that Kandahar was on the brink of falling, a senior local official in the city told the Guardian: “There is heavy street-to-street fighting in the heart of the city. The Taliban has almost captured Kandahar. As far as I know, only the governor’s compound is now in government hands. The city could fall in hours.”

Images on social media showed Taliban fighters near Martyrs Square in the centre of the Kandahar, barely 24 hours after the group overran the city’s central prison, releasing about 1,000 prisoners.

The fall of Kandahar – from where the Taliban launched their first insurgency in 1996 before rapidly taking over the country, and which served as the group’s capital until 2001 – would be hugely symbolic.

In Herat, insurgents took a number of key areas of the city, including the police headquarters as loud explosions and gunfire could be heard across the city, with heavy clashes centred on the governor’s mansion. Taliban fighters rushed past the Great Mosque in the historic city – which dates to 500BC and was once a spoil of Alexander the Great – and seized government buildings.

The fall of Kandahar and Herat, with the main approaches to Kabul already in the insurgents hands following the fall of Ghazni and Pul-e-Khumri this week, appears to signal the prospect of a countrywide defeat for the Afghan government.

Authorities in Kabul have in effect lost most of northern and western Afghanistan and are left holding a scattered archipelago of contested cities also dangerously at risk. The gateways to the Kabul have been choked with people fleeing violence elsewhere in the country this week, a western security source said, adding it was hard to tell whether Taliban fighters were also getting through.

As the battle intensified, the head of the British armed forces said a dangerous “security vacuum” risked opening up in the country, potentially enabling international terrorism to take a grip once again. Gen Sir Nick Carter said the best the Afghan government could hope now was a “military stalemate” that would enable it to negotiate a political compromise with the militants. “I think we have already got a humanitarian tragedy. The question now is whether it gets worse or not,” he told the BBC.

The former UK Foreign Office and defence minister Tobias Ellwood, who chairs the Commons defence committee, suggested it was not too late to stop the Taliban’s advance.

“It is actually not too late. To contain the Taliban – armed with AK-47s, landmines, RPGs – does not require a formidable force but it does require intelligence gathering, air superiority and strategy, which an international contingent could bring. It’s not too late for Britain to show that leadership.

“But we need to recognise the consequences of handing back a country to the very insurgency we went in to defeat and the terrorists that were involved in 9/11. The consequence of that will be further attacks in the west and a migration problem that makes the problem off Dover look miniscule.”

The week long onslaught represents a stunning collapse of Afghan forces and renews questions about the more than $830bn spent by the US Department of Defense on fighting, training those troops and reconstruction efforts – especially as Taliban fighters ride on American-made Humvees and pickup trucks with M16s slung across their shoulders.

The latest advances follow the Taliban’s capture of Ghazni on Wednesday night, 90 miles (145km) south of the capital and which sits on Highway 1 connecting Kabul and Kandahar.

The group occupied Ghazni’s government agency headquarters on Thursday after heavy clashes, a security official said. “The Taliban have completely captured Ghazni and control the city,” a senior local official confirmed to the Guardian by phone. “They broke into the prison and released around 400 inmates.

“They started their assault on the city at around 12am, and entered the city from several directions at around 2am.

“There were heavy street-to-street clashes between security forces and the Taliban. At around 8am they took over most key parts of the city and the city completely fell 30 minutes ago.”