Al Jazeera's James Bays, reporting from Helmand, said the Taliban had put out a statement saying "this is going to be a shameful defeat for international forces".
He said Brigadier-General Larry Nicholson, the commander of US marines in the province, avoided sounding upbeat about the operation, while cautioning that "there were tough days ahead".
Nicholson 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", our correspondent said.
Nato-led and Afghan forces have clashed with Taliban fighters in Marjah and the nearby district of Nad Ali in the continuing offensive that also seeks to eradicate drug traffickers that have controlled the region for years.
Three more Afghan civilians have died during the offensive, taking the total killed since the start of the operation to 15, Nato forces said.
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, US soldiers 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 Taliban fighters, who reportedly 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.
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 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.
As the fighting entered its fifth day, Pakistani intelligence officials said a US missile strike had killed at least three people near the Afghan border.
The two officials, who spoke on condition of anonymity because they were not authorised to speak publicly to the media, said Wednesday's attack destroyed a home in Tabbi Tool Khel village of North Waziristan in Pakistan.
The CIA has targeted homes, vehicles and suspected al-Qaeda hideouts in Pakistan's volatile North and South Waziristan tribal regions since December when a bomber killed seven of its employees in Afghanistan.