Katherine Guest was about to get the bus from Portsman Way in the town when she was turned away with her nine-week old son.
The 22-year-old, from Wenlock Rise, claims she was told by the bus driver he couldn't accept her onto the bus due to safety.
She says the driver on the 436 Arriva service to Shrewsbury was worried about her son 'choking to death'.
The mother-of-two was forced to walk to the doctors, where she ended up arriving late.
Following the incident, Arriva say it is investigating the incident but insist that breastfeeding mothers are welcome aboard its buses.
Ms Guest said: "I was breastfeeding my nine-week-old son when the driver said he couldn't accept me onto the bus because it was a safety issue.
"He was worried about my son choking to death on his bus. I was in too much of a hurry to argue. I was shocked that he had said that, so I didn't bother getting on.
"I had to walk down to the doctors. Where I am there's not many buses, so I didn't have any choice.
"I was angry. This shouldn't be a problem. It gets drilled into you, breastfeeding should be accepted and then something like this happens.
"I've never had a problem like this before. I haven't encountered issue when I've been breastfeeding in the past."
In 2010, Arriva Midlands signed up to a scheme supporting mothers who wish to breastfeed in public.
Ms Guest wants to get the word out there that breastfeeding should be acceptable.
She added: "I just want breastfeeding to be more talked about.
"I think Arriva need to talk to their drivers and put something into place where breastfeeding can be accepted on their buses.
"I want to get the word out there that it should be acceptable. There are tons of pubs and shops in Bridgnorth that have breastfeeding friendly stickers in the windows. I can go anywhere in the High Street and I don't get funny looks or anything."
Jamie Crowsley, general manager of the Shropshire area at Arriva Midlands, said: "We are currently looking into this incident and unfortunately we cannot comment upon the individual circumstances until the investigation has been completed.
"Our drivers do have a duty to ensure the safety of our passengers as they get on and off our buses and during their journeys and are mindful of this responsibility at all times.
"We would like to reassure breastfeeding mothers that they are welcome aboard Arriva buses and we will remind our drivers of this in light of the current investigation. We are seeking further details and when the report has been completed we review it to determine whether any lessons need to be learned and appropriate steps will be taken."