Whitchurch Road Surgery in the Heath area of the city currently has about 80 appointments a day for the approximate 8,000 patients on their list and has an estimated four-week wait for routine appointments.
The surgery has said some staff are doing overtime every day to see as many patients as quickly as possible.
"You have patient after patient coming in and the first five minutes is spent apologising for the fact they can't get an appointment," said Dr Rebecca Towner.
The 44-year-old has said she could see 30 patients in a morning, which is beyond the safe daily limit of 25 suggested by the British Medical Association (BMA).
The BMA said some GPs were seeing as many as 40 patients in a day due to an increased workload as some surgeries claim they cannot afford more doctors.
"Every single day you worry you're going to make a mistake," added Dr Towner.
"There's more risk with seeing patient after patient and less time to think. We're firefighting and it's exhausting."
She has admitted the situation can be "demoralising" and doctors she knows have considered quitting multiple times.
"I can't imagine doing anything else other than being a GP, but it's hard to see how we can carry on as it is at the moment," added Dr Towner.
Bosses at Whitchurch Road Surgery have put a cap on extra work "when possible" and claim a lack of funding because of the way the NHS pays GP surgeries means they have been unable to pay the same wages as other practices.
They say that means they are unable to hire new staff which has added to their workload and subsequently increased appointment wait time for patients.