Should I tell my professor of my crippling anxiety? It's really hard to admit, but I have a crippling anxiety when it comes to criticism from people of authority, especially university professors (at least I think so). This is currently negatively impacting my performance writing my thesis, as I would rather postpone indefinitely sending my thesis supervisor anything that could be wrong than face criticism, which makes me look lazy. Especially if I have encountered something that I don't know how to approach, I'd rather keep looking for answers myself for an impossible amount of time than ask for help (as I'm irrationally scared that I should be in a position to finish said task, since it was assigned to me).

I doubt that there is anything he can do, and I don't want to impose or require special treatment, but I feel like I have to be honest about why I'm not meeting deadlines as of late. My problem is that I don't know how to remedy this issue, and even if he knows about it, it's not magically going to stop impeding my work.

In addition, in case that I get anything along the lines of "I don't care", I'm pretty sure that I won't be able to handle it and I really don't want to give up my degree.
Your advisor is responsible for you, for good and for bad. This also means that if you have problems that are not of "technical" nature, that he still has to deal with it. It is in the advisors interest as well to find a solution for this, as he wants you to successfully finish. Hence, if you trust your advisor well enough to talk about this, then I would definitely recommend to talk to him. Even if you think he cannot help you, he should be still aware of the situation. Maybe he knows a way how he can at least make it a little easier for you.

Independently of this, I would suggest you to seek professional help. If you are at the level where such a problem has negative effect on your life and you don't see how you can solve it on your own or together with friends, then asking a psychologist is the right thing to do. Unlike what people commonly believe, they do not usually deal with "mad" people, but mostly with people who struggle with this or that problem in their lives, that they cannot find a solution for.

Though a word of advice: a psychologist is a person that you have to trust. You will be talking of problems that you might not talk about even with close friends. Hence, if you have even the smallest doubt that you can trust the psychologist you have chosen, look for another one. Do this, until you find someone whom you can fully trust and talk about everything that you want. And under no circumstances be afraid what the psychologist might think of you when you switch to another one. This is part of the game and everyone in this line of work knows it.