Is your advisor allowed to read your emails? In a typical workplace (North America), it is accepted that all communications through your work email are not personal property, and can be accessed by managers, IT department etc. Is this also true for graduate students? Assuming the student works in an university or research institute, the IT department obviously will have access to all emails. Does this also hold for advisors/supervisors? Are they allowed to, or typically have access to student emails?
What is the method of the email access?  If it is provided through the university or accessed on a university-provided computer, then it may be the case that they have a legal right to those contents.  This was the case at at least one of my workplaces in the USA.  If you are using your own email address, and your own computer, then I don't see how they could have the rights to access it.

Either way, it is best to have a separate email for all personal/private communication rather than risk it on your school email.