In Batman 1989, how did Bruce know Joker's identity before investigating?
I'm trying to establish some in-universe logic for Batman 1989. When Bruce Wayne gets home after witnessing the Joker and his goons kill Vinnie and his men in front of City Hall, Bruce tells Alfred he just found out Jack Napier is still alive and running Grissom's men. First of all, how did Bruce know the Joker was Jack Napier (who by all accounts should be dead)?
Along with the disfigured features from the plastic surgery, Jack is covered in makeup with green hair, and Bruce wasn't very close to him. In universe, does his new look conceal his identity, or not?
The following scene showing the TV news coverage doesn't mention that it was Jack Napier who committed the act, and he is consistently called the Joker by the media for the rest of the movie, so it seems the media doesn't know Jack is the Joker.
And along those lines, how did Bruce make the leap that Jack is running Grissom's men? Did he also assume the goons were Grissom's men dressed up like mimes?