In Batman Returns Bruce Wayne/Batman starts investigating Oswald Cobblepot / Penguin almost as soon as Oswald comes into the public light. Two scenes later he's investigating the potential link with the "Red Triangle Circus". Even when Alfred asks him why he's so suspicious he doesn't answer.
But while we, the viewers, all know the truth because we are privy to information that Bruce is not, what is it that tips Bruce off? I can't remember any moment that would have made him suspicious.