Taliban forces advanced on the capital of the volatile southern Afghan province of Helmand on Tuesday amid fierce fighting with government forces that threatened to cut off a major highway and prompted many families to flee.

The fighting near the town of Lashkar Gah comes three weeks after the Taliban won its biggest victory in the 14-year war, capturing the northern town of Kunduz and holding the city center for three days before government forces regained control.

"Helmand's capital appears to be under serious military pressure," a Western official said. "We're hearing reports about civilians fleeing in large numbers."

Helmand province is one of the world's biggest centers of opium cultivation, with a complex mix of warring tribal groups and Taliban insurgents creating a chronic problem for the Western-backed government.

Provincial Governor Mirza Khan Rahimi said heavy fighting had been going on for two days in the district of Gereshk to the north of the city. The fighting has threatened Highway One, the main transport artery linking the major southern city of Kandahar with Herat.

Farhad Dawary, head of the local Civil Societies Union, which represents non-government social organizations, said that after days of fighting, families were both fleeing to Lashkar Gah from outlying areas and trying to escape from the city.

"There is fear among the people in Lashkar Gah, there are lots of rumors the city might fall," he said.

Government officials said Lashkar Gah would not fall but one security source said the town of Baba Ji, to the north of the provincial capital had fallen and the Taliban itself said it was making major advances.

"The fighting is ongoing in full force and we have killed 25 government forces and seized a number of weapons," said spokesman Qari Yousuf Ahmadi.

KUNDUZ SHOCK

The fall of Kunduz at the end of last month came as a profound shock in Afghanistan, piling pressure on the government of President Ashraf Ghani and raising questions over the capacity of security forces to maintain stability.

Afghan government forces, which have taken on the bulk of fighting since international troops ended most combat operations last year, have struggled to contain the Taliban, which has stretched the army in a series of operations across the country.

As the battle continued near Lashkar Gah, there were also reports of heavy fighting in the Ghurian district near the major western city of Herat.

The Taliban raised their flag in the main district center for a few hours before police reinforcements arrived and cleared them out, Herat police chief, General Abdul Majid Rozi said.

As the fighting flared, Ghani chaired a meeting of the National Security Council and ordered military forces to step up combat operations against the insurgency.

Having briefly taken Kunduz at the end of September, the Taliban last week threatened the city of Ghazni, southwest of Kabul, cutting off the main route between Kabul and Kandahar for three days.

Earlier this week, they were also reported to have overrun the district of Ghormach in Faryab province on the northern border with Turkmenistan.

The deteriorating security situation prompted U.S. President Barack Obama last week to delay plans to pull American forces out of the country next year, with at least 5,500 troops now due to remain after 2017.

(Additional reporting by Krista Mahr, Hamid Shalizi and Aimal Yaqubi in Kabul, Jalil Ahmad Razaee in Herat; Writing by James Mackenzie; Editing by Alison Williams)