Is this normal or should I resign from this job...?

There seems to be two seperate issues here going on

1- Conflict over how you do your job

2- The mistake of you taking his hat.

The job is of a supervisor IS to supervise his/her staff, that's part of the job bud, sorry, but your going to run into this no matter which job you work at, its up to you to learn how to better cope with criticism. 

I'm not at your company, so I don't know what your job entails, nor do I know for sure if you are doing your job correctly/poorly. Having said that, its clear its not up to your managers standards, and since he is your superior, this is a big time issue that needs to be addressed. 

I wouldn't "mouth off to him" per say, because the last thing you want is to be fired, but you do need to speak up to him however! When he says things like "you should be able to do this_____ or _____" that's when you need to say "instead of being snarky, can you please tell me what it is you feel I am doing wrong? This way I can work on improving in those areas that you have listed". This gives him a chance to come back with areas that he feels you are doing poor at, and it gives you a chance to alter your way of doing things to his standards.

As for the hat thing? I would just apologize and move on.

Here's the thing about the real world- You are not going to get along with everyone, and people are going to have issues with you in turn, its how it works. As I have said, you need to find a way to deal with him as your manager, if he still keeps on you despite you doing a good job, that's when you report him to his supervisor.

typically normal. i work as a line cook many sous chefs/head chefs i worked for are very arrogant due to status/title but they lack basic skills like learning how to talk to employee's for the most part instead of treating them like shet, learning to follow up with employee's skills meaning if you have trouble with certain tasks still and worked there for a long time then they would retrain you at some places i worked for they would actually demote the people to dish washers or servers. 

trust me i worked for a LOT of sous chefs in particular who have huge ego's like this but fail at their own job. you need to understand employee's/people have rights as well and you can't just treat people as you please you could get fired/demoted for doing so. 

if your supervisor gave any fk's about the company he worked for or the staff he should be supervising he should of been retraining you. as for the hat thing sounds just like some cry baby bs it's not a big deal but he crying over spilled milk. 

i wouldn't talk back though they can use that against you and fire you. just suck it up do your job and move on. i've gotten yelled at many times by sous chefs/head chefs over dumb things not worth stressing over. you need to remember they can fire you doesn't matter if your clausterphobic or what not you need to adjust. i worked around quite a few restaurants scared as shet not knowing wtf im doing but like any other job over time it gets better. my current job is the only one that will take longer to adjust because i am unaware of what position i am really doing as i was hired as a prep cook but they make me a line cook instead with prep cook pay or something. plus the menu changes daily and i switch positions on what im doing almost daily so it's just weird it's not like a regular line cook job where i stick to 1 station learn it then move to the next this one got me all over the place with little directions from the sous chefs so i just do whatever they ask me to do and try my best. they dont like it then they can fk off and find a better system 

You need to take the hint. Get better at your job.

How have you been there 4 years without being good at it?

I worked for 4-5 months part time and I was offered a promotion. I took it because I felt sure that more raises & another promotion would come and they did. I got regular raises because I was under the standard store manager pay scale despite going a great job.

It took me 14 months from not knowing how to do anything to being a general manager.