Updated at 7:42 a.m. Eastern.
International and Afghan troops have begun a key combat phase against insurgents in southern Afghanistan and expect heavy fighting, officials said Monday, in an operation that is crucial to the U.S. strategy to turn around the nine-year war against the Taliban.
The allied forces were moving into two or three areas around Kandahar city in southern Afghanistan at once to pressure the Taliban "so they don't get the chance to run away," Shah Mohammad Ahmadi, chief of Arghandab district northwest of the city, said Monday.
"Before, when we have tried to get rid of the Taliban, when we cleaned one area we found more Taliban in a different one," he said.
At least 16 Americans have already died in the new offensive, which began to ramp up more than a week ago, reports CBS News correspondent Mandy Clark.
Some of the fiercest fighting has been in Zahri district, a strategic location for insurgents. Not only can they mount attacks on Kandahar city from Zahri, but also control the intersecting highway -- the main supply route between Kandahar and Helmand.
Meanwhile, a years-long fight, just as deadly as the battle for Kandahar may be, continues for U.S. troops operating out of tiny, remote bases dotted along the frontline of the war; Afghanistan's border with Pakistan.
"The fight there is brutal, the intensity unlike anything we've witnessed in nine years of covering this war," reports CBS News correspondent Lara Logan, who spent weeks experiencing the frontline battles with American troops and a "60 Minutes" crew.
U.S. soldiers from the 101st Airborne Division are locked in a never-ending battle with an enemy that uses the border as an open door, says Logan.
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A top NATO officer said Sunday that the alliance a few days ago had launched its "kinetic," or combat, phase of "Operation Dragon Strike," a joint military push with Afghan forces around Kandahar intended to rid the area of insurgents and interrupt their ability to move freely and stage attacks.
"It is a significant ground operation with air support," German Brig. Gen. Josef Blotz, a NATO spokesman, said at a news conference. "We expect heavy fighting."
"Afghanistan and coalition forces are destroying Taliban positions so they will have nowhere to hide," Blotz said. "Once this is done, insurgents will be forced to leave the area or fight and be killed."
NATO said militants have fought back with rocket-propelled grenades and small-arms fire.
It said no Afghan or coalition troops have been killed in the operation.
The push in Kandahar is seen as key to the Obama administration's strategy to turn around the nine-year war as insurgents undermine the ability of an Afghan government to rule much of the country. Kandahar remains particularly dangerous; seven U.S. troops have been killed in Kandahar this month. Another three have been killed in the south, but no further details have been released.
"First, they are clearing the area of Taliban and then they are searching the area for mines," said Kareem Jan, district chief of Zhari, west of Kandahar.
In another volatile section of the nation, British officials said Monday that they were in contact with Afghan authorities about the disappearance of a British aid worker and three of her Afghan colleagues. The four were ambushed Sunday as they traveled in two vehicles in northeastern Kunar province in eastern Afghanistan. Police fought a gun battle with the kidnappers near the attack site before the assailants fled, Kunar police chief Khalilullah Zaiyi said.
Tim Waite, a spokesman for the British Embassy in Kabul, said officials were working closely with all relevant local authorities and said the worker's family had been contacted.
Steven O'Connor, communications director for Development Alternatives Inc., a global consulting company based in the Washington, D.C., area, said late Sunday its employees, including a British national, were involved. The company works on projects for the U.S. Agency for International Development in Afghanistan.
Britain's Foreign Office in London said it could "confirm that a British national has been abducted in Afghanistan."
Also in the east, NATO confirmed an air strike following a cross-border attack on an Afghan National Security Force outpost in Khost province, near Pakistan, on Friday killed more than 30 insurgents. A follow-up on Saturday resulted in several more insurgents killed. It said there were no NATO casualties.
It said that insurgents were attacked under the rules of engagement in the cross-border skirmish. It did not give further details. It is unusual for NATO helicopters to attack targets on the Pakistan side of the border, where some of the casualties are believed to have occurred.