He added: "The Taliban have left the areas, but the threat from IEDs [improvised explosive devices] remains."
Marjah resident Haji Mohammed Jan told the BBC the Taliban had tried to stop people leaving, but he and others had managed to escape.
"All we had we have left behind. We don't like fighting. We are tired of it."
Dawud Ahmadi, a spokesman for Helmand's governor, said nearly 1,000 displaced families had arrived in the provincial capital, Lashkar Gah.
Afghan MP Fauzia Koffi told the BBC that residents were confused about the operation, because the military offensive was very different from the suggestion at a recent conference in London that elements of the Taliban should be engaged in talks.
She also said that the Taliban had not allowed civilians to leave Marjah and if civilian casualties increased it would adversely affect the opinion of the Afghan people.
The BBC's Frank Gardner, at Kandahar air base, says a clear difference is emerging between Nad Ali - where British troops are operating, and Marjah to the south where US troops are focused.
General Stanley McChrystal on ''protecting the people of Afghanistan''
While British forces have been able to move quickly to their objectives, US Marines are advancing slowly and painstakingly, being held back by snipers and more home-made bombs than they had expected, he said.
The BBC's Ian Pannell, embedded with British forces in Nad Ali, says troops have been spreading out to try to reassure locals, listen to their complaints and assuage their fears.
He says the locals appeared happy to see the Afghan army but did not want the arrival of police, who they see as corrupt, partisan and inept.
Military intelligence experts believe most Taliban who have chosen to stay and fight are concentrated around Marjah.
In northern Marjah on Monday, an armoured column came under fire from at least three sniper teams, AP news agency reported.
HIMARS ROCKET SYSTEM
US artillery rocket system developed by Lockheed Martin
Carries six rockets on mobile launcher. Range can exceed 45km (28 miles)
Alternative payload is one long-range ATMS guided missile
Uses include targeting artillery, air defences, light armour and in infantry support
Able to launch and move quickly before retaliatory strike
Prototypes used during Operation Iraqi Freedom. Entered full US service in 2005
Reuters quoted US Marines as saying they had twice unsuccessfully tried to clear one bazaar area in Marjah of enemy positions.
Despite the setbacks, Nato and Afghan commanders insist they now have enough troops to hold the ground taken and will soon be bringing in hundreds of newly trained police to re-establish Afghan government control.
Operation Moshtarak, meaning "together" in the Dari language, is the biggest coalition attack since the Taliban fell in 2001.
The operation is also considered the first big test of US President Barack Obama's new "surge" strategy for Afghanistan.
Nato has stressed that the safety of civilians in the areas targeted is their highest priority.
However, on Sunday two rockets fired from the High Mobility Artillery Rocket System (Himars) in the Marjah area hit a building and killed 12 Afghan civilians.
In the second incident on Monday, Nato said the five Afghan civilians killed were mistakenly believed to be insurgents planting explosives on a road in the Zhari district of Kandahar province.
Gen McChrystal immediately suspended all use of the Himars rocket system and Afghan President Hamid Karzai ordered an investigation.
However, on Monday a senior Nato official told the BBC the rockets were not off target as initially reported.
The building was targeted as coalition forces were receiving fire from it, the official said, but they did not realise there were civilians inside. Two insurgents along with the 12 civilians were killed, the official said.
At a news conference on Monday, Gen McChrystal said that before the operation had begun, President Karzai had stressed the importance of protecting Afghan civilians.
"This operation has been done with that in mind," he said, adding he was "exceptionally proud" of how coalition forces were performing.
Speaking at the same conference, Afghan interior minister Mohammad Hanif Atmar urged Taliban fighters to lay down their arms and take up the government's offer of reconciliation.
"There is no way you can win there, the Afghan people are determined to win," he said.
The 15,000-strong coalition force includes 4,000 US Marines, a similar number of British troops plus a large Afghan contingent. Soldiers from Canada, Denmark and Estonia are also involved.
Two Nato deaths related to Operation Moshtarak have so far been confirmed.
On Saturday, a British soldier, Lance Sergeant David Greenhalgh of 1st Battalion Grenadier Guards, died in an improvised explosive device (IED) attack, while a US soldier was killed by gunfire in Marjah.
Another British casualty was announced on Monday, although not connected to Operation Moshtarak. UK officials said the soldier from 2nd Battalion, The Duke of Lancaster's Regiment died during an night-time patrol near Musa Qaleh in Helmand on Sunday.
At least 20 Taliban fighters were killed and another 11 detained on Saturday, the Afghan army said.
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