The bombing is set to continue
Mr Bush said al-Qaeda and the Taleban should not underestimate America's resolve.
"They think they can run, they think they can hide, because they think this country's soft and impatient," he said.
"But they're going to continue to learn the terrible lesson that says don't mess with America."
America's new envoy to Afghanistan, Zalmay Khalilzad, said the bombing would continue as long as a threat to the US remained.
Meanwhile, the Americans are questioning their most senior Taleban prisoner - the former ambassador to Pakistan, Mullah Abdul Salam Zaeef - after he was refused asylum by the Pakistani Government.
Mullah Zaeef, the best-known face of the deposed Afghan leadership, was taken into US custody on Saturday - while the main American targets for arrest, Bin Laden and his main ally, Taleban supreme leader Mullah Mohammad Omar, have so far eluded capture.
The US has also announced that it has taken into custody one of the highest-ranking members of Bin Laden's al-Qaeda network, Ibn Al-Shaykh al-Libi, who reportedly ran training camps for the group which is accused of carrying out the 11 September attacks on the US.
US military officials indicated that Mullah Zaeef had been taken on board the USS Bataan in the Arabian Sea. More than 300 prisoners are currently being held by the American military on the Bataan and in the southern Afghan city of Kandahar.
Mullah Zaeef gave regular news conferences broadcast around the world during last year's US-led campaign in Afghanistan.
It is not clear whether the Pakistan authorities delivered him directly into the hands of US forces.
Omar's escape
Taleban leader Mullah Omar escaped from his hideout in southern Afghanistan on a motorbike, a senior Afghan intelligence official said on Saturday.
Mullah Omar was reported to have left Baghran with four of his supporters.
Afghan foreign ministry spokesman Omar Samad said that the Taleban leader would be "captured dead or alive".
In Germany, the authorities say a man they arrested on suspicion suspected member of Bin Laden's network on Saturday had no link to the organisation.
Heinz Jurgen Vitz, a prosecutor for the city of the western city of Moenchengladbach, said the man remained in custody on a charge of carrying false papers, but there was no evidence he had anything to do with al-Qaeda.