American troops will remain under control of their commanders while on duty in Afghanistan despite calls from Afghanistan President Hamid Karzai (search) to have more authority over them.

"Of course, our troops will respond to U.S. commanders," Bush said with Karzai standing at his side at the White House during a press conference on Monday. At the same time, Bush said the relationship between Washington and Kabul (search) is "to cooperate and consult" on military operations.

Karzai stressed the need to keep a U.S. military force there to keep his country on the road toward stability.

"The United States has been a country with whose help, we have rebuilt our country, are in the process of rebuilding our country, and you have been at the forefront of that effort with us in Afghanistan and the rest of the world," Karzai told Bush.

He added that a strategic partnership is needed because even though assembly and parliamentary elections will be held there in three months, "we would like the world to recognize that ... Afghanistan will not suddenly stand on its own feet."

The private meeting with Bush Monday was an opportunity for Karzai to vent some grievances over U.S. involvement in his country's struggle to recover from decades of instability.

But during a new briefing with the U.S. president, Karzai stressed the Afghan people's tremendous gratitude for the United States' help in getting their country back on its feet.

"Thank you for all you've done for Afghanistan. We are very, very happy. We are grateful," Karzai told Bush at the White House press conference.

For his part, Bush held up Afghanistan (search) as a model of emerging democracy and anti-terror partner.

"I am impressed by the progress you're making toward a market economy and a full-fledged democracy," Bush told his Afghan counterpart, urging transparency in government and the economy and noting that women are increasingly taking a larger participatory role in government and society. "I've got great faith in the future of Afghanistan."

At Karzai's Oval Office session with Bush, the centerpiece of a four-day U.S. visit and the two leaders' first such get-together since September, the Afghan leader hoped to win a commitment for a long-term — perhaps permanent — U.S. military presence in his country. But Karzai also said in advance of the meeting at the White House that he wants greater control over American military operations there.

Relationship of 'Cooperate and Consult'

Approximately 20,000 U.S. troops are in Afghanistan, and no end is in sight to their mission — including the still unfruitful search for Al Qaeda leader Usama bin Laden (search). That is in addition to about 8,200 troops from NATO countries in Kabul and elsewhere. But U.S. receptiveness has been limited to the idea of a rigid, permanent arrangement there.

Bush and Karzai signed a memorandum of understanding establishing a "strategic" relationship between the two countries that involved high-level exchanges between officials on political and economic interests. Karzai said the long-term deal would ensure that his country continues to receive reconstruction assistance and training from U.S. military and police.

Karzai also said that he wants to take over custody of the hundreds of Afghans detained in military jails in Afghanistan and at Guantanamo Bay, Cuba (search), during and after the 2001 U.S. invasion that ousted the repressive Taliban regime.

"Our policy, as you know, is to work our way through those who are held in Guantanamo and hand them back over to their countries," Bush said Monday. "We will do this over time. We have to make sure the facilities are there."

And, citing reports of prisoner abuse by American forces at the U.S. military's main base at Bagram, Karzai said over the weekend that he wants promises of punishment for any U.S. troops guilty of mistreatment.

But the Afghan leader stressed that he does not hold the American people as a whole, accountable for the actions of a few in such abuse situations.

"We are of course sad about that but let me make sure that you all know that that does not reflect on the American people," he said, again stressing that "the Afghan people are grateful, very grateful to the American people" for helping free their country.

Bush has promised that advancing freedom's march across the globe is the top foreign policy goal of his second term. So the meeting with Karzai was a chance to showcase his administration's successes in the War on Terror and support for young democracies.

Bush has said Afghanistan's new constitution, elected president and upcoming parliamentary elections in September represent "remarkable progress." But the protests and other difficulties in Afghanistan show the complexities involved in turning chaotic, poor countries with long histories of violence into stable, thriving democracies.

The Poppy Problem

Karzai began his U.S. stay by sharply denying a reported State Department cable that said he has not worked strongly enough to curtail production of opium (search), the raw material for heroin. The cable, from the U.S. Embassy in Kabul to Secretary of State Condoleezza Rice, said the U.S. crackdown there has not been very effective, in part because Karzai "has been unwilling to assert strong leadership," The New York Times reported Sunday.

"The reduction of poppies is actually because the Afghan people refrained themselves. A little has been eradicated by force. The rest is people refraining themselves from growing the poppies. So that's a tremendous sign; that's a great sign of hope. And if you go forward with proper alternative livelihoods, people will no longer grow poppies," Karzai told FOX News on Sunday.

Production of opium has soared to record levels since the fall of the Taliban — roughly 5,000 metric tons of opium was produced there in 2004 — leading to warnings that the former Al Qaeda haven is fast turning into a "narco-state." Last year, cultivation yielded nearly 90 percent of the world's supply.

Bush said on Monday that he and Karzai discussed the poppy issue and stressed the need to rid that country of the crop.

"There's tooo much poppy cultivation in Afghanistan," the American president said. "I made it very clear to the president that we have got to work together to eradicate poppy crops."

Karzai said his country is well on the road to eradicating the crop, since poppy cultivation is undermining Afghanistan's economy and "giving us a bad name."

Recent anti-American protests across Afghanistan killed at least 15 people and threatened a security crisis for Karzai's feeble central government.

The White House blamed a Newsweek report — later retracted by the magazine — for igniting the violence. The May 9 story said Pentagon investigators had found evidence that interrogators at Guantanamo placed copies of the Koran, the Muslim holy book, in washrooms to unsettle suspects, and flushed one down a toilet.

But Karzai has blamed opponents of his ties with the United States and of his reconciliation efforts with the Taliban.

"There's a respect, there's this freedom in America for religion, and there are Muslims, on a daily basis, praying in mosques in America. And there are Korans, holy Korans, all over America — in homes and mosques. So it was a political act against Afghanistan's stability, which we have condemned, which the Afghan people have condemned," he said.

Karzai was also meeting with Rice, Defense Secretary Donald Rumsfeld, congressional lawmakers and the new head of the World Bank, Paul Wolfowitz. Karzai received an honorary degree Sunday from Boston University and will pick up another at the University of Nebraska-Omaha.

FOX News' Liza Porteus and The Associated Press contributed to this report.