To become apparent - when you realise after a certain amount of time, or through new information, that something is, or is going to be a certain way.
e.g. As she became more and more tired, it became apparent she was not going to win the race.
To turn out - this is used to discuss an outcome, and usually the outcome is a surprise or unexpected.
e.g. Although she seemed very tired during the race, it turns out she won.