...and unknowingly, ended up in the hands of the Crime Doctor in a couple issues of Detective Comics!
The Crime Doctor Calls At Midnight
Batman is dealing with a new villain in Gotham, the Crime Doctor, who uses medical practices to help criminals. During the course of this case, Batman get injured, and Alfred, instead of being able to go to Bruce's usual doctor, has to go to Dr. Bradford Thorne, who stitches up Bruce Wayne's wound. Bruce Wayne then invites Thorne to a special event at the Wayne Foundation. But...a case calls them both away. Batman tries to stop the criminals, but is confronted by the Crime Doctor, who sees the special bandage he put on Bruce Wayne.....
.....allowing Crime Doctor Thorne to figure out that Bruce Wayne is Batman! Then, the criminals figure they won't have to pay the Crime Doctor if they kill him, so trap Batman and the Crime Doctor in a lab with a bomb!
Murder In Quicksilver
Batman saves the Crime Doctor from the bomb, but escapes an exhausted Batman. The criminals go to their leader, crimelord Sterling Silversmith (whom Batman has faced before). The criminals reveal they knew that the Crime Doctor knew Batman's identity (but didn't reveal it). Sterling Silversmith has the Crime Doctor brought to him, but Thorne won't reveal the name to Silversmith as it would violate normal patient/doctor confidentiality. So, Silversmith poisons Thorne with mercury....but Batman does save him (tracking down Silversmith by his thugs).
Batman was too late, with Thorne remaining in a coma, so Batman doesn't know if Thorne revealed his identity to anyone else. Thorne must have recovered, at least temporarily, as the Crime Doctor was spotted in an issue of Crisis On Infinite Earths.
The Crime Clinic
Batman and Robin got to work stopping him, and did, even with the doctor working to save patients.
The Crime Doctor returned in Batman #18 (August-September, 1943), escaping prison, and even shooting Robin, but died in that issue after being shot by one of his patients.