PDO Login/Register system review
Here are several tips:


Use OOP - this way you could start the session only once
Hash your passwords - this way no one can steal it from the db, or at least the chance is lower
Use MVC - separate your HTML from the PHP code
Move your db connect credentials to the .ini file - this way it could be easily changed and .ini files can be cached by the server


Feel free to check my code for user authorization here in my pet project.