Military aircraft struck Taliban positions close to the district's main town, Mingora, early on Monday.
There were also reports that the army had jammed an FM radio station run by the Taliban in Swat.
"It was through those FM stations that they were able to pass their information and go on a campaign of their own, a propaganda campaign," Kamal Hyder, Al Jazeera's correspondent in Peshawar, said.
Rehman Malik, Pakistan's interior minister, said on Monday that up to 700 fighters had been killed in the offensive so far.
He pledged that the operation would "continue until the last" fighter has been ousted.
But Haji Muslim Khan, a Pakistani Taliban spokesman, told Al Jazeera that the military was "lying" about the number of dead Taliban fighters.
"They simply want to impress the Obama administration, because that's where they get their money from," he said.
"The operation in Swat is being carried out at the behest of the American administration."
The US has expressed support for the offensive, after calling on Pakistan to do more to root out Taliban fighters in Pakistan who are said to plot attacks on American and Nato forces fighting in Afghanistan.
The ongoing fighting has displaced hundreds of thousands of people, in an exodus that is said to be the largest movement of people in Pakistan in 60 years.
About half a million people are believed to be leaving the region, in addition to more than 550,000 people already displaced from Swat and nearby areas since August.
Abdur Rahman, one displaced resident who fled to a refugee camp in the nearby town of Mardan, told Al Jazeera: "People from all over - from Matta, Mingora and from everywhere - [are fleeing] on foot. Women and children and even old women and old men.
"Some of them died on the road, but no one was willing to offer us any help - neither the army nor the Taliban.
"They are both committing atrocities and cruelty against the ordinary people."
It was not possible for Al Jazeera to verify accusations of atrocities.
The army lifted a curfew on the area briefly on Sunday to allow residents to flee, before imposing it again with a "shoot on sight" order for anyone who violated it.
Many of those fleeing have headed to nearby villages, but thousands more are gathering at any of 11 refugee camps around the province.
Reporting from Islamabad, the Pakistani capital, Al Jazeera's Sohail Rahman said: "That exodus of civilians continues ... it's a desperate situation.
"Those that have managed to escape the Swat valley during that curfew period on Sunday are certainly making their way to those camps in and around the southern part of the region, and outside Peshawar."
Al Jazeera's Hyder said what is of real concern now is the "harrowing tales we have been hearing from people trapped" inside Swat.
"They are begging for a corridor to be able to get out, but so far they are still under curfew," he said.
"There is no electricity, there is no food. If they are not given help on time, many of them are going to start dying. There was no [government] action plan to deal with this emergency and this crisis."