But within hours of Bush's speech in Alabama on Monday, at least three people were killed when a bomb exploded outside a hotel used by Iranian pilgrims in the Iraqi city of Karbala.
Earlier, a bomb blast northeast of Baghdad near the town of Baquba left one Iraqi dead and 15 wounded.
Elsewhere, a large explosion echoed across Baghdad, followed by mortar fire near the headquarters of the US-led administration in Iraq. There were no reports of casualties.
The US president's comments were his first since 16 US soldiers were killed when resistance guerrillas shot down their CH-47 Chinook helicopter on Sunday.
"The enemy in Iraq believes America will run. That's why they're willing to kill innocent civilians, relief workers, coalition troops. America will never run," Bush said, despite falling approval ratings in the US over the war.
But he did not refer to the downing of the helicopter or three other deaths which made Sunday the second worst day overall for the Americans of the entire conflict. At least 20 US soldiers were wounded.
Wounded US soldiers arrive at
Ramstein airbase in Germany
Since the invasion, at least 250 US soldiers have been killed by hostile fire. The number of Iraqis killed since March is in the thousands and steadily climbing.
At least three people were killed when a bomb exploded on Monday outside a hotel used by Iranian pilgrims in the holy Shia city of Karbala, 90km south of Baghdad, Shia officials said.
They said the bomb appeared to have been planted in a nearby car and had destroyed much of the front of the hotel.
The target of the earlier explosion near Baquba was the deputy governor of the northern Diyala province Aqil al-Hamid. He escaped uninjured.
Al-Hamid told AFP the explosion on Monday morning narrowly missed the vehicle he was travelling in. One civilian was killed.
Guerrilla resistance is stiffening, in contrast to the crushing of Iraq's regular army seven months ago - a rout partly caused by Saddam's delusion the invasion was only a feint, his former deputy prime minister was quoted as saying.
As US troops and Humvees guarded the helicopter wreckage on Monday near the flashpoint town of Falluja, west of the capital, villagers from old men to children celebrated the attack, Reuters reported.
They called it the perfect present to mark the Muslim holy month of Ramadan. The region where the helicopter was downed is a hotbed of anti-US anger.
"We usually celebrate Ramadan at the end of the month. Now we are celebrating in the beginning after these infidel Americans were shot down," said taxi driver Abd Allah Hissein.