The two-day Camp David meeting comes as a number of issues plague the U.S. effort to support Karzai's government, eliminate the Islamist Taliban that supported the Sept. 11, 2001 plot and find al-Qaida leader Osama bin Laden.
Afghanistan is experiencing the most violence since U.S.-led military action ousted the ruling Taliban in 2001. Besides detonating roadside and suicide bombs, the Taliban took 23 South Korean church volunteers hostage and killed at least two of them. The Taliban has demanded the release of its imprisoned members, saying otherwise more hostages will be killed.
Presidents Bush and Karzai said Monday that there would be no exchange of prisoners for hostages.
The leaders vowed Monday to destroy the Taliban, which Karzai described as a defeated and frustrated force that must resort to attacking civilians.
President Bush reassured Karzai that the United States is committed to helping Afghanistan eliminate the radical Islamic group, which he called a challenge to the free world.
"The real question is whether or not those of us who have the blessings of liberty will continue to pursue policies -- foreign policy, security policy -- aimed at not only protecting our homeland, but aimed at laying a condition for peace to prevail," Mr. Bush said at a news conference with the Afghan leader.
Presidents Bush and Karzai disagreed about the influence of Iran on Afghanistan's problems, however. While Karzai called neighboring Iran a partner in the fight against terrorism and narcotics, Mr. Bush said he would be cautious about calling Iranian influence in Afghanistan positive.
The United States has accused Iran of supplying insurgents in Afghanistan and Iraq with weapons and of trying to develop nuclear weapons.
