It was the deadliest assault on American forces in Afghanistan in nearly two months, officials said on Sunday.
The troops were part of a major operation on Sunday against fighters who have vowed to subvert legislative elections on 18 September - the next step towards democracy after a quarter-century of fighting.
Rebels also stepped up attacks elsewhere, wounding two US Embassy staff in a roadside bombing in the capital and killing a senior pro-government cleric as well as four policemen in separate assaults in the country's south.
Though the US military offensive has left dozens of suspected rebels dead or captured, a number of American troops have also been killed, including 13 this month.
US and Afghan officials have warned the violence may worsen in the run-up to the polls.
The bomb tied to the bottom of the wooden bridge exploded on Sunday as the last of three Humvees was slowly crossing it, said Bashir Ahmad Khan, the government chief in Zabul province's Daychopan district.
"The American vehicle was tossed into the air and off the bridge. It's totally destroyed, as is the bridge," he told The Associated Press.
The three wounded troops were hit by shrapnel from secondary explosions from inside the burning Humvee as they tried to pull the four out of the vehicle, the military statement said.
At least 13 US soldiers have died
in Afghanistan in August alone
The three were evacuated to a nearby base and were in stable condition.
Major-General Jason Kamiya, the US-led coalition's operational commander, said the blast would "strengthen ... the resolve" of the troops to safeguard the polls.
It was the deadliest attack on American forces since 28 June, when 19 service members were killed in eastern Kunar province when a Navy SEAL team was ambushed and a helicopter shot down.
In violence elsewhere, a roadside bomb exploded near a convoy of US Embassy vehicles on the outskirts of the capital, Kabul, lightly wounding two American staff members, embassy spokesman Lou Fintor said.
In neighbouring Kandahar province, armed men riding a motorbike killed cleric Mawlawi Abdullah - the latest in a string of attacks on religious leaders who have openly condemned the Taliban and other extremists.
Abdullah - a senior figure in the Islamic Ulama Council - and a colleague were killed as they walked out of a mosque after praying at dawn, said Interior Ministry official Dad Mohammed Rasa.
Violence may surge in the run-up
to the 18 September elections
Police set up roadblocks around the area, but no one was arrested.
Armed men also targeted a senior cleric in southern Spin Boldak district on Sunday, but he fled into his house when the assailants opened fire and was unhurt, local police chief Abdul Wasai Khan said.
In the eastern province of Kunar, rebels ambushed two tanker trucks hauling fuel to a US military base, burning the vehicles but letting the drivers go, officials said.