"The main aims of the operations are to extend security throughout the areas so as to allow the local population to enjoy a normal life and take part in the forthcoming elections free from intimidation and violence," a British military statement said.
At least 15 Taliban fighters have been killed in the Gereshk district, Afghan officials said.
Elsewhere, at least 14 mine-clearing personnel working for the United Nations were abducted in the eastern Paktika province, officials said.
The US marine operation - dubbed Strike of the Sword - is aimed at seizing control of the lower Helmand valley, the world's largest opium producing region, from the Taliban.
Al Jazeera's Zeina Khodr, reporting from Lashkar Gah, said: "There has been little resistance in this US offensive in the south of Helmand but the Taliban can move from one village to another and most of them are a part of the civilian population so it is very difficult.
"They [Taliban] do not engage coalition and Afghan forces in direct combat. Their tactics are suicide bombings, creating insecurity, ambushes, roadside bombs," she said.
However, several hundreds US marines have been engaged in some heavy fighting over the past few days and many areas have been covered in mines, military commanders said.
The US forces, backed by the Afghan national army, moved into Baramshar on Saturday after taking the main districts of Nawa and Garamsir, as well as Khananshid, the previous day, local officials said.
Michael Griffin, a UK-based author and expert on Afghanistan, said that the Taliban were not standing up and fighting because they could not possibly take on the size of force they were facing in Helmand province.
"The strategy of the new head of US forces in Afghanistan, Stanley McChrystal, is to eradicate an effective Taliban presence in Helmad ... by taking, holding and introducing some degree of stability to areas previously held by the Taliban," he said.
"This is looking like it is working very well, but the problem with the Taliban is always that when they retreat before a superior force they are going to pop up somewhere else."
Military officials have said that once the about 4,000 US marines, backed by a 650-strong group of Afghan soldiers, have taken areas previously held by the Taliban they will stay and secure them rather than pulling out as has been the policy before.
Al Jazeera's Josh Rushing, reporting from a US camp in Helmand province, said one of the biggest challenges facing the marines was the heat.
"Even in the tents it is extremely hot, and there's no where to go to escape the nature of the heat," he said.
"The troops in villages like Nawa have it much worse because they're suppose to carry all the food and water they need so they don't have to be resupplied for up to six days," he said.
"A marine officer told me they've had a number of heat injuries although not life-threatening yet, but it's definitely one of their major concerns."
Barack Obama, the US president, had ordered 21,000 additional troops to Afghanistan and expects the total number of US forces to reach 68,000 by the end of 2009 in support of the offensive that started on Thursday.
The Afghan deminers, who were employed by the Mine Detection and Dog Centre (MDC), were taken as they worked in the eastern Paktika province.
|US marines are being supported by hundreds
of Afghan security personnel [Reuters]
"Fourteen workers of MDC demining organisation who were busy serving the Afghan nation were kidnapped by unknown gunmen in Tandan area of Gardez pass," Zemarai Bashary, the interior ministry spokesman, said.
Azizullah Wardak, Paktika's police chief, said they were seized as they were travelling towards Khost province.
While the Taliban is known to operate in the area, Wardak could not confirm if they were responsible for the kidnapping, but he did criticise the deminers for going into the area without informing the police.
However, Bashary said that three men had been arrested over the case and efforts were under way to secure the deminers' release.
Afghanistan is one of the most heavily mined countries in the world, and the increase in violence amid a thriving Taliban insurgency has slowed clearance work.
Two-thirds of the country's mines have been cleared over the past two decades with the rest expected to be removed by 2013, but experts fear Afghanistan can no longer meet that goal because of increased fighting and a drop in international funding.