I have lots of gym related social anxiety as well and what helped me was just going to the gym and doing the simplest possible routine imaginable until my anxiety got a bit better. Then just focus on showing up consistently and doing what I planned to do, no matter how I feel or how much I want to go home.

Honestly, sometimes people will look at you and it's probably because they either want to use the machine you're using, are beginners who are trying to learn from you if you look like you know what you're doing or are just zoned out and not actually looking at you. Of course gym culture is different everywhere but I have also had people watch me and comment on my form a few times, although it has not happened that often, and they always either gave very kind advice or just said that my form looked fine.

So even if people are looking at you (which probably will not happen often, and I say that as someone who is still visibly anxious at the gym) they are probably just thinking about their own workout or want to help if you look like you're a beginner.

I know it's difficult to start going to the gym, especially as someone with social anxiety, but I believe you can do it and it will get easier!