Bessie was never shown to travel inside the TARDIS on the show.
Bessie appeared only infrequently, but most of her episodes were during the Third Doctor's era, when the Doctor's use of the TARDIS was heavily restricted. Bessie was a way to get around his lack of mobility during this time1, and her appearances declined dramatically after The Three Doctors, when full use of the TARDIS was restored.
Following that episode, Bessie's few appearances were in short driving scenes, and she was never shown to interact with the TARDIS at all. After the Fourth Doctor left UNIT, Bessie remained under their care on Earth, and was put in mothballs:
Doctor: If there's an accident. We'd better stop any engagement. Have you got a helicopter available?
Brigadier: Oh, better than that, Doctor.
[Outside Gore Crow Hotel]
(Soldiers take a tarpaulin off a bright yellow roadster with the number plate WHO7.)
Doctor: Ah ha! Bessie!
Brigadier: Well, I knew how fond of it you were, so when you last went on your travels I had it put in mothballs.
Doctor Who Season 26 Episode 1: "Battlefield"
1 It didn't hurt that Jon Pertwee had a thing for classic cars