US Defence Secretary Robert Gates has warned that â€œhard fightingâ€ lies ahead, in his first visit to Afghanistan since the launch of a major offensive there.
After meeting military chiefs overseeing the anti-Taliban operation in southern Helmand province, Mr Gates also said some progress had been made.
Preparations have already begun to secure control of neighbouring Kandahar province, military commanders said.
Additional troops ordered by US President Obama have begun arriving.
About 6,000 of the 30,000 extra forces assigned to Afghanistan have already arrived in Afghanistan. Thousands more are due to arrive over the next few months.
But Mr Gates warned: â€œPeople still need to understand there is some very hard fighting, very hard days ahead.â€
The offensive in Helmand, targeting the Taliban stronghold of Marjah, has been described as the biggest operation since the invasion of Afghanistan in 2001 – it involves Nato, US and Afghan troops.
But officials have recently hinted that the current action in Marjah is a â€œpreludeâ€ to a bigger operation.
Nato commander Gen Stanley McChrystal has made it clear that Kandahar is the next priority for troops, once enough reinforcements have arrived.
The general said that, although the district was not under Taliban control, it was â€œunder a menacing Taliban presence, particularly in the districts around itâ€.
The BBCâ€™s Chris Morris in Kabul says that, as was the case with Marjah, international commanders are making little effort to conceal plans about where they intend to take the fight to the Taliban.
There is a vast swathe of territory across southern Kandahar and Helmand provinces from which forces want to drive the Taliban before re-establishing a functioning civilian infrastructure, our correspondent says.
But, he adds, military operations are deeply unpopular with local people and military commanders are aware of the need to get the balance right.
Afghan police and government agencies have already started to deploy in and around Marjah but officials warn that the region is not yet totally free from Taliban influence.
On Monday Mr Gates discussed the progress of the operation with Afghan President Hamid Karzai.
On Sunday President Karzai visited the former Taliban stronghold of Marjah for the first time since the beginning of the offensive in Helmand.
He promised elders that the town would be rebuilt and appealed to local people for support.