You shouldn't have to "westernise" yourself if you're a good candidate. Be confident in yourself - wearing a turban doesn't mean you're less educated or not as good at the job as a white person. Try not to let your nerves show in your interviews, and while your parents are only thinking about your future, you just need to reassure them that you know you have the right skills and personality to get through an interview.
A lot of Muslim girls who wear the hijab don't shake hands with male interviewers, and I guess that is one thing that can give them a disadvantage in an interview. Not everyone is aware that it's because of religious reasons that they're not shaking hands, and it can come across as rude sometimes, especially because a lot of them refuse it quite suddenly and don't actually explain it. So if you or anyone else reading this thread has an issue like this in interviews i.e things you can't participate in because of religious reasons, then you should make it clear or find an alternative rather than outright refusing to do something and being blunt. I just think that's the main downfall for a lot of Muslim girls and I've seen it happen a lot at graduation ceremonies too where the girl just walks straight past, whereas there are others who actually do manage to make it clear that they won't be shaking hands without actually seeming rude about it.