The victims -- which include two women -- were targeted by angry mobs over the rumours, spread mostly on Facebook, that said human heads were required for the massive $3 billion project, police chief Javed Patwary said.
"We have analysed every single case of these eight killings. Those who were killed by lynching mobs -- no one was a child kidnapper," Patwary told reporters in Dhaka.
More than 30 other people have been attacked in connection with the rumours.
Patwary said police stations across the country had been ordered to crack down on rumours, and at least 25 YouTube channels, 60 Facebook pages and 10 websites have been shut down.
AFP has identified several posts still on Facebook that share the rumour, however.
Mob lynchings are common in Bangladesh, but the latest incidents are particularly brutal.
Local media said they started after reports circulated of a young man allegedly found carrying the severed head of a child in the northern district of Netrokona.
Among the latest victims was a single mother-of-two, Taslima Begum, who was beaten to death in front of a Dhaka school on Saturday by a mob which suspected her of being a child kidnapper, a police official told AFP.
A deaf man was also beaten to death outside the capital that day while trying to visit his daughter.
Police said eight people have been arrested over Begum's murder, and at least five others detained for their role in spreading the rumour on social media.
- Loudspeaker pleas -
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Police are so concerned about the deadly fallout in rural towns that officers are trying to counter the web rumour using loudspeakers.
"We are building awareness about the rumour and ask people not to get panicked," a police chief in northwestern Chapainawabganj district said.
Some 6.1 million Ansar paramilitary security forces and village guards have also been asked to warn villagers, according to media reports quoting Ansar major general Kazi Sharif Kaikobad.
Meanwhile beggars fearful of being lynched were wearing their identity cards to prove they were not strangers to a particular area, local media reported.
The lynchings could be "a sign of people's distrust in the existing law and order system", Dhaka University sociology professor Monirul Islam told AFP.
But he did not rule out the possibility that some people were deliberately trying to trigger panic or unrest in the community.
The bridge -- which is set to be Bangladesh's biggest -- is being built on the Padma, a major tributary of the Ganges.
Rumours of human sacrifices being required for a bridge in Bangladesh have surfaced before, with several people attacked in 2010 over another structure, according to local media.