A local charity said it had registered 2,241 families displaced by the fighting.
Pakistan officials have said that about 50 suspected Taliban fighters and eight paramilitary troops have been killed in the operation so far.
The paramilitary Frontier Corps shelled the Maidan area of Lower Dir overnight into Tuesday, a senior military officer said.
"We destroyed several militant hideouts in heavy artillery-shelling of suspected bases in the area," the officer said, on condition of anonymity.
The government's assessment comes a day after a source told Al Jazeera's Kamal Hyder that thousands of civilians were streaming out of Lower Dir due to the army's "accurate" and "spread out" assault.
Sufi Muhammad, a religious leader who has been holding peace talks with the central government, has suspended dialogue with Islamabad as a result of the army assault.
"We are suspending talks with the government until the military operation in Dir is halted," Aamir Izat, a spokesman for Muhammad, said on Monday.
The Lower Dir district, which lies in the Malakand division of the North-West Frontier province, is close to the border with Afghanistan.
The military action in Lower Dir comes weeks after the government allowed the Taliban to implement their interpretation of sharia (Islamic law) in the neighbouring Swat valley, after talks brokered by Muhammad.
Prior to Muhammad's move on Monday to suspend negotiations with Islamabad, a spokesman for President Asif Ali Zardari said that the government would fulfil its pledge to allow the establishment of Islamic courts in the northwest of the country.
But he said that the government, which allowed sharia in Swat in an attempt to limit violent activity in the region, would not permit the fighters to spread their area of influence.
In recent days, fighters from Swat began entering another district, Buner, which lies just 100km from Islamabad, the Pakistani capital.
Later reports suggested that the fighters had begun to pull out after military action was threatened.
Critics of Pakistan's deal with fighters in Swat say that it has only emboldened the Taliban.
The US in recent days has increased pressure on Pakistan to confront fighters on its soil.
General David Petraeus, the head of US central command, said Pakistan's leaders should focus on the looming threat posed by fighters within their borders.
"The most important, most pressing threat to the very existence of their country is the threat posed by the internal extremists and groups such as the Taliban and the syndicated extremists," he said.
Across Pakistan, more than 1,800 people have been killed in a wave of al-Qaeda and Taliban-linked attacks since July 2007.