Bush said, with Karzai standing at his side at the White House: "Of course, our troops will respond to US commanders". At the same time, Bush said the relationship between Washington and Kabul is "to cooperate and consult" on military operations.
The two leaders met on Monday at the White House amid new tensions caused by the ongoing US military role in Afghanistan, 3-1/2 years after US-led forces ousted the Taliban from power following the 11 September 2001 attacks.
The two leaders went out of their way to present a picture of harmony at a news conference in the White House East Room, and a tangible outcome of their talks was a joint strategic partnership they signed aimed at strengthening ties.
There are about 20,000 US troops in Afghanistan, costing about $1 billion a month.
That is in addition to about 8200 troops from Nato countries in Kabul and elsewhere.
Bush said that Afghan prisoners under US control in Guantanamo Bay, Cuba, and elsewhere, would be slowly returned to their home countries.
"We will do this over time," he said. "We have to make sure the facilities are there."
Bush had high praise for Karzai as a valued anti-terror partner and credited the Afghan leader with "showing countries in the neighbourhood what's possible".
Karzai thanked Bush for helping to put his country on the path to democracy. But he also came to their meeting with a long list of grievances.
Karzai wants more control over US military operations in his country, custody of Afghan prisoners held by the United States and more assistance in fighting the opium trade.
Karzai had requested more
control of US military operations
Karzai commented on recent reports of abuse of Afghan prisoners by their American captors.
"We are of course sad about that," he said, speaking in fluent English. But, he added, "It does not reflect on the American people."
Similarly, a report - later retracted - in Newsweek magazine earlier this month that alleged desecration of the Quran by US forces does not reflect American values, Karzai said.
While claiming the original report was not responsible journalism, Karzai said: "Newsweek's story is not America's story. That's what we understand in Afghanistan." Karzai noted that many thousands of Muslims are going on a daily basis to mosques in America, without incident.
Bush and Karzai pledged to work more closely together amid continued instability and protests in Afghanistan. "It's important for the Afghan people to understand that we have a strategic vision about our relationship with Afghanistan," Bush said.
Opium cultivation has risen
since the US invasion in 2001
He said the United States and Afghanistan had signed a "strategic partnership" that establishes "regular high-level exchanges on ... economic issues of mutual interest".
"We will consult with Afghanistan if it perceives its territorial integrity, independence or security is at risk," Bush said.
Karzai said that he hoped Afghanistan would be free of opium poppy crops within five to six years and that Afghan farmers could find alternative crops such as honeydew melons and pomegranates.
Opium cultivation rocketed
Opium poppies are the raw material for heroin.
Their cultivation has rocketed since the fall of the Taliban in 2001. Last year, cultivation reached a record 323,700 acres (131,000 hectares), yielding an estimated 80% of world supply.
"Afghanistan is suffering from the cultivation of poppies, which is undermining our economy," Karzai said. "It's giving us a bad name, worst of all."