I gotta add my two cents to the OP and the pro-sports guy (further down). I went through the same kind of inner battle at first... afraid of what other people would think, how'd I'd be viewed, what would my friends and family think, etc etc etc. I finally concluded four things:
A) I have every right to seek out and find what I enjoy. The same as everyone else. My preferences are just as valid and important as every other person's and so are yours
If we all collectively hide it, the stereotypes will never change. The only way to improve the future is to stand tall and honor yourself and what you like. Don't be ashamed of it.
C) Those that are close to you and genuinely care about you, aren't going to care or ridicule you for it. They may be surprised at first and have questions. They may be curious. Expect the questions and realize they are just curious and not a putdown.
D) It's all a matter of how you present it. I'm sure you've heard the saying "He could sell and air conditioner to an Eskimo" - it's all a matter of how you present it. For example, I never say "I love fat girls" to the outside world. Fat has such a negative connotation that I pretty much avoid the word altogether. What I'll say is something like "I know as guys we're all expected (or taught) to like skinny girls but I've just never found it attractive. I like girls that I can rough house with and not be afraid of breaking" or "I prefer the curves and softness of a little bit bigger girl" or "I'm like my girls like I like my steak: Thick and juicy" - especially if said with a grin and slight chuckle, people respond positively to that. I was very surprised at the response I got from other others. ESPECIALLY girls. Even skinny ones were enthusiastically grateful to know that such a guy actually exists. I had guys, that were previously ragging on fat girls amongst a group of guys, come up to me later after finding out I prefer bigger girls, and talk to me about it saying things like "I'm glad you mentioned that, I thought I was the only one". One guy told me "You've seen my wife. That's no accident" as he gave a little grin and walked away.
The key is, as others have said, is to be confident about it. If you're insecure, and apologetic and ashamed of it, then people will feel like you're doing something wrong or you're "weird". If you're confident and 'own' it and you embrace it, then people will go "Oh, that's different but to each their own" or "We all have our thing, that's his" or "Not for me, but more power to you". I'm sure you get the drift. I can speak from experience on that. I am so happy and grateful with myself for deciding to express my preferences when opportunities arose instead of hiding them with embarrassment. I have literally had a positive impact on many people over the last few years.
You mentioned and anxiety issue. I can't help you much there. That's a different issue altogether. I imagine there are books and other material that might be helpful in resolving that within yourself. I do believe one way of approaching that might be to start small and with someone insignificant in your life. A casual coworker that you rarely see, or a classmate, if at school, that you only share one class with or whatever. You don't have to pour your heart out, but start real small. If he points out a skinny girl and says "Man, look at her!" You can say something like "Yeah, she's got a cute face but kinda skinny for my taste. I do love long hair on a girl though." Conversations like that.... very minor and insignificant with someone irrelevant and unimportant to your life. If you feel like they 'reject you' then it's no big loss. I would encourage you to think about it and try it out. Little baby steps, one at a time. You'll probably be surprised just how often you *don't* get a bad response
To the pro-sports guy: You're social status puts you in a position of, at least some, influence and power. Not that you're obligated to, but if you were to openly date a larger girl that you really liked and loved and who was okay with the media attention, even if negative, you could be a positive inspiration to many and be one more crack in the crumbling stereotype. A great example to others. Again, it's your choice obviously but just like has been mentioned both by me and by others, if you're confident and have a good relationship and people see you two together as a happy couple, the public at large will respect that and be happy for you two, even if a little critical at first. It's the end game that matters, not the opening play