Why cant Edward read Bella's mind in the twilight series?

It's her special gift. Like Edward was able to understand people's thoughts when he was human, Bella is immune to a vampire's abilities.

Well it's actually because Bella have a private mind. Nothing and nobody can affect her mind (except for Edward at the end of Breaking Dawn.) When Bella turns into a vampire, her power is that she can shield people, including herself, from attacks to the mind.

I guess you could say that. It's actually because when Bella was a human, she had some type of mental shield/layer inside her body. In breaking dawn, her vampire gift is a force-field type thing. When she uses her force-field that is the mental shield she had when she was human. So when she uses that shield, her mental layer becomes exposed. So if it is exposed, Edward and Aro can read her mind, and Jane can cause her pain.