Amid fresh questions on Osama bin Laden's whereabouts, Mr. Bush said he expected bin Laden's al Qaeda network to try to regroup and strike again.
Still, he pledged, "We will defeat the threats against our country and the civilized world."
Mr. Bush used an address to cadets at Virginia Military Institute to give an update on his war against terrorism at a time when dubious results of his Middle East peace initiative and the lingering, elusive specter of bin Laden have claimed the headlines.
The president said Secretary of State Colin Powell made progress toward Middle East peace on his just-completed mission, which ended without an Israeli-Palestinian cease-fire.
"I want to thank Secretary Powell for his hard work at a difficult task. He returns home having made progress toward peace," the president said.
Powell returns from his Middle East tour having failed to obtain a cease-fire or persuade Israel to complete a pullout from Palestinian areas despite Mr. Bush's April 4 demand that a full withdrawal be done quickly.
Mr. Bush again called on Israel to continue its withdrawal from the West Bank, and he said the Palestinian Authority must act on its words of condemnation against terrorism, reports CBS News Correspondent Mark Knoller.
Mr. Bush also called on friendly Arab regimes to stop praising suicide bombers, and to stop providing money to their families.
He declared: "All parties must say clearly a murderer is not a martyr."
Mr. Bush said resolving the Israeli-Palestinian crisis is part of his broader — and long — battle against terrorism.
"Wherever global terrorism threatens the civilized world, we and our friends and allies will respond, and respond decisively," he said.
Despite signs of wavering by Arab partners, Mr. Bush proclaimed the international coalition behind his anti-terror efforts "strong and united and acting." Without naming Iraq, he renewed his determination to squash the threat of outlaw regimes in an "axis of evil."
He put a humanitarian face, too, on his cause: "America has a much greater purpose than just eliminating threats and containing resentment because we believe in the dignity and value of every individual. America seeks hope and opportunity for all people in all cultures."
Mr. Bush counted off the accomplishments in the first phase of the war in Afghanistan, where bin Laden and his al Qaeda network and Taliban allies were rousted by U.S. military might.
"We're clearing minefields, we're rebuilding roads, we're improving medical care and we'll work to help Afghanistan develop an economy that can feed its people without feeding the world's demands for drugs," Mr. Bush said.
But a new videotape surfaced in Qatar this week showing bin Laden alive and at work to terrorize Americans. It remains unclear when the tape was made, but serves as vivid reminder that the Bush administration does not know where the suspected mastermind of the Sept. 11 attacks is. A White House official said the government's assumption is that bin Laden was present during December's fierce firefighting in Tora Bora, Afghanistan, but escaped.
Mr. Bush focused instead on the one senior bin Laden lieutenant U.S. forces have nabbed, Abu Zubaydah, the third-ranking figure in the al Qaeda terror network.
"He's not plotting and he's not planning any more. He's under lock and key; and we're going to give him some company. We're hunting down the killers one by one," Mr. Bush told thousands of cheering military cadets in formal dress uniforms.
Citing intelligence gathered from the laptop computers, maps and drawings seized as U.S. forces scoured al Qaeda hideouts in Afghanistan, Mr. Bush gave a chilling forecast of the potential for more attacks.
"As the spring thaw comes, we expect cells of trained killers to try to regroup, to murder, to create mayhem and try to undermine Afghanistan's efforts to build a lasting peace," the president said.
After the speech, Mr. Bush was returning to the White House to meet with Lebanon's prime minister, Rafiq Hariri, on ways to prevent skirmishing along Lebanon's southern border with Israel from merging with the West Bank bloodshed into a wider Arab-Israeli war.
Mr. Bush hit a snag on the return trip when Marine One shut down, for reasons not immediately clear, and he had to cross the university lawn to a backup helicopter.
Advisers said his VMI speech, which made him the first president to visit the state university since 1964, was not meant to break new ground in the anti-terrorism campaign, only to sustain support for the long campaign by letting Americans know how it is going.
The president is likely to use a planned address at West Point for another such update in several weeks, said another White House aide.
Public approval of Mr. Bush's military campaign against bin Laden's network and terrorist targets in Afghanistan remains very strong, along with the president's overall job approval rating, ranging from 75 percent to 80 percent.
Public opinion polls show far more mixed sentiment on the Middle East, where Americans strongly support Mr. Bush's recent steps to intervene and mediate peaceful settlement, but doubt they will be successful. The most recent surveys found Americans just about evenly split on whether the Bush administration has a clear and well thought-out policy on the Middle East.