How to tell my supervisor that I have severe phobia of flying? The phobia of flying is discouraging me to travel for conferences, although quite a number of them are very good opportunities for me to "market" the researches of our lab and of mine. Apparently, I can't overcome this phobia even with the help of therapists. I almost passed out due to panic attacks on the airplanes, and after the flights, I always felt too exhausted to do anything for an entire week. 

I think it's time to let my supervisor know that I can't to go to any conferences that I have to fly there. How can I do it in the most professional way?

Thanks
Since it doesn't sound like it's likely that you will be able able to work around it in the immediate future, I think you should schedule a meeting and just have the conversation. Explain your phobia, explain the effects, and explain that you have worked with therapists and that the problem is unresolved.

Treat it like any other health or mental health disability that will affect your ability to carry out the tasks graduate students normally do and make it clear that you will attend conferences by train or bus where possible. There are many questions about health and disability on this site and the consensus generally is that if it will impact your performance, you should bring it up with your supervisor and colleagues as soon as possible.  Since this has already become an issue, you should do that now.