The usual choice is an interior privacy lock: it locks from inside with a thumbturn or a push button, without a key, and in an emergency it can be opened from outside.
On a bathroom it is almost essential that it can be opened from outside. If a child gets locked in or someone feels unwell inside, the door does not have to be broken. Many privacy locks have a slot on the outside that can be turned with a coin.
For a bedroom, it depends on how it is used. If it is just for privacy, the same lock will do. If you want to lock it with a key, for example in a room you rent out or a room where you keep things, an interior lock with a key is fitted.
We also replace privacy locks that are broken or do not lock properly. It is all in interior door locks.