The 112-year-old bridge, which is currently closed to road traffic during the day, got stuck in the open position last Thursday, meaning people on the east side, many elderly, faced a long walk back to the other side of town.
Cllr Phil Trumper tweeted that it was causing chaos and that investment was needed in a replacement with better hydraulics.
After another breakdown yesterday (Mon), he told the Gazette: "We should be looking now at replacing it, at least in the next 10 years.
"It seems to be breaking down more often.
"It's over 100 years old and was designed when we still had the horse and cart - it's not fit for the 21st Century.
"It's nearly a mile walk into town by going over the new bridge and it's affecting businesses.
"You just can't bring a town centre to a standstill.
"The issue is the age and you can't just keep putting bandages on these things."
He said investment in a replacement bridge,which ideally would have wider footpaths for pedestrians, would also help to resolve issues of increased traffic at the Spital Bridge junction.
Cllr Trumper's tweet won support with Richard Dowson adding: "We definitely do need a new bridge and it needs to be wider. There is room both sides of the river to the northern side of the bridge to make it wider."
Yesterday (Sep 20), John Field added: "And stuck again today. Doing nothing is becoming an expensive option."
"Every breakdown brings Whitby to a standstill.
"There is no practical alternative for thousands of pedestrians," she said.
"We need it working reliably.
"If that can’t be achieved with piecemeal repairs to the existing bridge, then we must see swift and effective investment from the county council to deliver a fully functioning crossing.
"When the current double leaf swing bridge was built, a temporary bridge was built alongside the site.
"That was needed to allow demolition of the old bridge and the construction of the new one.
"Sadly, there’s no way that could be done in 2021 for the total cost in 1909, which was £22,582 14s 4d!"
The bridge was constructed in 1908 and was opened the following year.