1. overly dramatize something, adopt a false attitude/appearance
put on a show of remorse/to impress someone
If you accuse someone of 'putting on a show' you are accusing them of overreacting or having a response which is much bigger than what would be expected or merited. If someone is 'putting on a show' it is similar to accusing them of being a 'drama queen' or 'making a spectacle of yourself' meaning looking for undue attention. This is an informal phrase.