After the on-site interview I visited the campus for an in-person interview 2 weeks ago.
They supported my flight and hotel.
I thought I was good at the interview and the director stated that she will be in touch after several weeks.
Is it okay if I send follow-up email?
I am so nervous and cannot focus on my work.
Any comments will be appreciated.
Unless they gave you a specific time frame, then yes, it is fine to ask them about the current state of your application. 

If you have other options, then keep them open until things are definite, but don't bring them up when asking for an update. 

Don't become a pest, but a couple of weeks is a pretty good interval.