A Taliban spokesman disputed the claim saying they have Nato forces under seige.
Al Jazeera's James Bays said the Taliban had put out a statement saying "this is going to be a shameful defeat for international forces".
Nato-led and Afghan forces have traded gunfire with Taliban fighters in Marjah and the nearby district of Nad Ali in the continuing offensive dubbed Operation Mushtarak.
Three more Afghan civilians have died during the assault in Helmand, taking the total killed since the start of the operation to 15, Nato forces said.
Meanwhile, the commander of British troops in southern Afghanistan said two missiles that struck a home on Sunday, killing 12 people, did hit their intended target.
It was originally thought the rockets missed their target.
Major-General Nick Carter's comments came as Afghan authorities handed over the bodies for burial. He said Taliban fighters were in the house.
Nato has said the aim of the operation is to re-establish Afghan government control in the area so security and civil services such as police stations, schools and clinics can be set up.
Hoping to avoid prolonged gun battles, marines called for long-range artillery support to disperse sniper squads delaying their advance into Marjah in the Taliban's southern heartland.
For the first time since the offensive started on Saturday, US forces fired non-lethal artillery "smoke shells'' in a bid to intimidate enemy fighters who also lobbed rockets and mortars at them.
"We are trying not to be decisively engaged so we can progress, but we're having some difficulty right now," Captain Joshua Winfrey, Lima company commander, said.
James Bays, Al Jazeera's correspondent in Helmand, said Brigadier General Larry Nicholson, the commander of US marines in the province, did not sound upbeat about the operation and had warned that "there were tough days ahead".
Bays said Nicholson had echoed "comments from the governor of Helmand, who ... believes there are significant numbers of Taliban still in some of these villages, he believes some of them are foreign fighters, among them Pakistanis, who are holding out in some of these areas".
Despite the continued firefights, marine officials said the resistance was more disorganised than in previous days.
"We're not seeing co-ordinated attacks like we did originally," Captain Abraham Sipe, a US army spokesman, said.
"We're still getting small-arms fire but it's sporadic, and hit-and-run tactics."
Nato has confirmed 15 civilian deaths since the start of the offensive, with the latest three deaths coming in three separate incidents.
The military alliance said in two of the incidents, Afghan men came toward Nato forces and ignored shouts and hand signals to stop. The troops then opened fire and killed them.
In the third incident, two Afghan men were caught in the crossfire between fighters and Nato forces.
Both were wounded, and one died of his injuries despite being given medical care, the Nato said.
On Sunday, two errant US missiles struck a house on the outskirts of Marjah, killing 12 people, half of them children.
Afghan officials say three Taliban fighters were inside the house.
Bays said there had been protests on Tuesday in Lashkar Gah, the provincial capital of Helmand, over Sunday's deaths.
He said: "This followed the arrival of the 12 bodies of the people who were killed. When they arrived, there were protests on the streets of the provincial capital.
"This is not what the military wanted, they wanted this to be a model operation with very limited casualties."
Caught in crossfire
An Afghan human rights group said on Tuesday that they had counted 19 civilians killed since the beginning of the operation.
Four of those were people who were caught in the crossfire when they left their homes, the organisation said.
"Their neighbours tell us that the bodies are outside and they want someone to pick them up. They say they're scared if they go outside they will also be shot dead"
the director of Afghan Rights Monitor
"Their neighbours tell us that the bodies are outside and they want someone to pick them up. They say they're scared if they go outside they will also be shot dead," Ajmal Samadi, the director of Afghanistan Rights Monitor, said.
It was unclear whether Nato or Taliban forces were to blame for the deaths, he said.
Nato officials have reported the deaths of only two international forces troops during the operation so far, with one American and one Briton killed on Saturday.
In Marjah, US Marine and Afghan squads have been skirting the booby-trapped streets of the town, where Taliban snipers have been firing from haystacks built over small canals.
The Marines' goal has been to link up with other companies that were airdropped into the town on Saturday, but progress has been slow.
Residents said they were scared to be seen with Nato forces.
One man, Wali Mohammad, said "don't take pictures or the Taliban will come back to kill me," as marines searched his compound.
Mohammad said he strongly suspected fighters would return to the area as soon as the marines moved on.
Elsewhere in Helmand province, Nato and Afghan forces killed more than 10 Taliban fighters as they chased a group of three vehicles in pursuit of a Taliban commander in Washir district, west of the area where the offensive is going on.
In a separate incident, unrelated to the Marjah offensive, a Nato airstrike in neighbouring Kandahar province killed five civilians and wounded two.
Nato said they were mistakenly believed to have been planting roadside bombs.
At least 1,240 families have fled the massive military onslaught against the Taliban, many sheltering with friends and relatives, Helmand authorities said Tuesday.
No camps have been set up for the displaced lest they become permanent structures, said Daud Ahmadi, spokesman for Mohammad Gulab Mangal, the Helmand governor.
"We deliberately did not give permission for the camps to be set up for the 1,240 families who are displaced because we did not want the camps to become permanent," Ahmadi said.
The refugees are "either living with their relatives or have rented houses for themselves," in the provincial capital Lashkar Gah he said, adding that a number of international agencies were giving assistance.
Lashkar Gar is only one destination for the displaced, and Ahmadi was unable to say how many people had fled elsewhere.