The clown mentioned the source of his scar twice in the film. His father was very angry because of his serious expression when he killed his mother, which caused this scar.

Moreover, he later confessed that his lover was disfigured by the gang to show that he would not abandon his lover, and he also bit the blade and finally scared his lover away.

When the correct explanation is not given in the film, and the clown's character is always portrayed in the film, then neither is credible.

In addition, the old Batman explicitly mentioned that the clown's disfigurement was a tragedy in the chemical plant accident.