Since I lived far from the gym, but LTH was just near, I needed to make my gym neccessities as compact as possible, so that they would fit in my handbag together with things I needed for school. And I succeeded quite well in doing just that - so let me show you how to significantly reduce the amount of things you carry.
On the overview picture, you can see my gym clothes and shoes and a hair bun maker - if you have long hair, you can make a top bun without any elastic, which is much less damaging for your hair. Also, if you go to the sauna, the ends will be hidden inside the bun, thus limiting their damage by the heat. Your hair will also get less greasy as it doesn't touch your sweaty back and shoulders.
My cosmetic bag - I generally went to the gym after school, so I didn't bother with bringing make-up and if I used skincare, it was all one-use samples I gathered from magazines and Korean cosmetic stores. Towel is one thing you can save space on, as you really don't need a duvet-sized one. Next to it is a lock - most places in Sweden do not provide locks, so you need to bring your own to saunas, pools, gyms etc. A code lock is better as you don't need to worry about loosing keys.
As for cosmetics, use travel-sized things. You can buy travel containers and transfer your cosmetics in them, or buy small-sized products and after finishing them, using them as travel containers (more sustainable). It does help that Korean stores give tons of small sizes of their products as gifts. As for removing make-up, make-up removing tissues are the best. Just take one (or, I just use 1/3 or 1/2 since using whole one is a waste) and put it in a small ziplock bag, e.g. from Q-tips. And you're 300 g down.
You can see my mini face wash (Beyond), deo, face mist (great if you need to freshen up quickly during breaks), mini lip balm, mini shower gel. I do not bring hair care as I only wash my hair at home, because I do not blow dry it and it would be too cold to go out with wet hair, but you could put your shampoo and conditioner in small bottles too. I couldn't understand girls that dragged some 2 kg of cosmetics to the gym.
Since I had access to free tea, coffee and syrup at the faculty room, I only put a couple of drops of fruit vinegar from Korea in my bottle and brought it empty and only filled it with water at the gym, to reduce the weight of my bag. If I stayed for more classes, I brought a soluble vitamin tablet in a small ziplock bag to but in the next batch. And of course fruit and Oatly (Swedish oat-based milk substitute) as snacks.
The most important thing ever - bathroom slippers. I've never had a fungi foot infection and I credit my almost obsessive use of bathroom slippers in all shared showers for that. Showers are wet, hot (= paradise for microbes causing infections) and if you check other people's feet, a great source of nasty stuff. Do your health a favor and get yourself a pair. It sometimes feels weird since most people don't care, but just because most people take the risk does not mean you should.