KANDAHAR, Afghanistan — NATO forces claimed yesterday to be on the verge of achieving their first major success against the Taliban since the launch of their campaign to pacify southern Afghanistan.

The militia’s fighters were reported to be scrambling to escape from an operation to crush them inside a corridor bounded by a road and a river west of the strategically vital city of Kandahar.

“There are definite signs that they have had enough,” a senior officer with the NATO-led International Security Assistance Force said. “There is no doubt they have taken a big hit.”

ISAF calculates that more than 500 Taliban have been killed in the 13 days since the launch of “Operation Medusa.” If true, that would mean that a third of the 1,500 fighters believed to be in the area had been wiped out. Seven ISAF soldiers and two Afghans died in the fighting.

Military sources insist that they have solid evidence gleaned from satellite imagery and reports on the ground that the statistics are accurate.

But ISAF officers were in no hurry to declare victory, saying that this was only the first stage in a long campaign.

“The word victory has a funny way of coming back and biting you,” one of them said. “What we are doing here is trying to establish a secure environment in which we can start the reconstruction and development work that is essential to bring real stability.”

The operation was led by the 2,200 Canadian troops in the area with support from the Americans and the Dutch. The ground assault was heavily reinforced with artillery and air attacks. Units of the Afghan army aided by international support teams also took part.

ISAF cited it as an example of what they are capable of doing by skilful use of their existing resources. But those directing the fight make it clear that the addition of the 1,000-strong fighting reserve that they are seeking from NATO states would significantly increase their ability to speed up results. “The key is sustaining the effort,” one senior officer said.

Troops moving into the battle area as the Taliban vacated found an intricate system of defenses and logistics. “There were bunkers, fall-back positions, weapons caches, and medical evacuation facilities,” one officer said. A factory for making roadside bombs was also discovered. The fields had been sown with mines, booby traps, and anti-personnel devices, which will make the return of the estimated 15,000 families who fled the area difficult.

The fundamentalist militia has been able to re-establish its control of parts of Kandahar and neighboring Helmand province in the five years since it was driven from power by the post-September 11 American-led invasion.

Few coalition troops were deployed in the area, and no serious attempt to deliver it to government authority occurred.

Taliban forces, most believed to have come in from Pakistan, have been building up their strength, often in alliance with local drug lords, and have hired extra gunmen from among the young unemployed.