"This is going to be a long struggle," Bush said at the Afghan presidential palace on Monday. "Ideological struggles take time."
He said: "Are there difficult days ahead? Absolutely. But are the conditions a lot better today in Afghanistan than they were in 2001? Unquestionably, undoubtedly they are better."
The talks with Karzai came after Bush addressed US soldiers at Bagram air base, which lies just outside Kabul.
'Continued assistance'
Karzai said that Afghanistan was grateful for the American help, saying its people "don't want to be a burden on the international community for ever", but he said that setting a timetable for the withdrawal of US forces was premature.
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"I am confident we will succeed in Afghanistan because our cause is just"
George Bush,
outgoing US president
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"Afghanistan will not allow the international community leaving before we are fully on our feet, before we are strong enough to defend our country, before we are powerful enough to have a good economy," he said.
Bush assured Karzai that Washington would provide continued support to the country after Barack Obama, the US president-elect, takes power on January 20.
Bush emphasised the need for more troops in Afghanistan, but said decisions on the number of soldiers sent to the country would be taken by Obama.
"I recognised that we needed more troops. President-elect Obama is going to make decisions on troops. And we've been calling on our Nato allies to put in more troops," Bush said.
Extra troops sought
There are about 31,000 US troops in Afghanistan but General David McKiernan, the senior US commander in Afghanistan, has asked for an extra 20,000 soldiers to be sent to the country.
Troops from the US and Nato have faced particular difficulties in providing security across southern Afghanistan, where fighters linked to the Taliban have been resurgent this year.
Obama has promised to make Afghanistan a higher priority, saying the Bush administration has been too distracted by the unpopular Iraq war to pay Afghanistan the attention it deserves.
Bush also said it was important for the US to keep working with Pakistan to maintain pressure on fighters along its border with Afghanistan.
"If Pakistan is a place from which people feel comfortable attacking infrastructure, citizens, troops, it's going to make it difficult to succeed in Afghanistan," he said.
"The more we can get Pakistan and Afghanistan to co-operate, the easier it will be to enforce that part of the border regions."
Khalid Pashtoon, an member of the Afghan parliament, said that Afghans are looking for a continued US commitment against the Taliban.
"Eight years ago, when there was the fall of the Taliban and the arrival of the international community to Afgahnistan. For eight years they the Taliban have lived in the tribal areas of Pakistan, from where they have attacked Afghanistan," he told Al Jazeera.
"The Taliban have created a great amount of disturbance against the Afghan people, so Afghans think that [to now] the US presence in Afghanistan did not bring any prosperity or peace.
"People here had lost their hope for the future. Now there is hope that the [forthcoming Obama] adminstration in the United States will bring peace."