Avon and Somerset Police received multiple calls about the van being driven dangerously on Wellsway in the south of the city at about 3.30pm on Friday.
The Ford Transit collided with a number of other cars before coming to a stop.
Members of the public blocked the vehicle with their cars and broke its windows in order to take the keys from the ignition, police said.
The driver was detained by officers, but an ambulance was called when they became concerned for his welfare.
He was taken to Royal United Hospital in Bath but died overnight, a spokesman for Avon and Somerset Police said.
His next of kin have been informed.
Avon and Somerset Police said it had alerted the Independent Office for Police Conduct because of the circumstances of the death, and said a mandatory referral would be made.
Detective Inspector Keith Smith said: “At this early stage of our inquiry it is not currently clear what happened and what led to the man sadly passing away.
“Our thoughts are with his family at this difficult time and we will carry out a thorough investigation to provide them with the answers they deserve.”
Mr Smith added: “While officers spoke to a number of people at the scene we believe there are others who witnessed the incident who we are keen to talk to.
“We’d also like to hear from anyone who drove on Wellsway, Churchill Bridge or Lower Bristol Road yesterday afternoon who has dashcam footage of the incident, as well as anyone who captured it on their phone or private CCTV.
“Similarly, if your vehicle or property has been damaged, please let us know.”
Anyone with any information is urged to contact police on 101 quoting reference number 5222143789.