No, a GIF isn't lossy. It's just 8bit.
A JPEG is lossy.
Also try throwing PNGs through an optimiser before using somewhere, to shave off extra bytes. Like PngOptimizer or optipng.
Reducing a PNG to 8bit should yield smaller images than GIF. (It's also a much fairer comparison too!) You may also find that doing the above and then saving it as 32bit could result in a smaller image.
Jpegs are already pretty compressed. you aren't gonna be getting much out of compressing them. However PNG's have a lot of room for optimization with various programs and techniques. None of the more useful methods are exactly "lossless" but the image basically looks the same afterwards (its usually reducing the amount of colors in the image).
Just search around for something that works really. I have a friend who uses this but i've never felt the need to use it. My shitpost folder is only 10 gigabytes compared to my 2.5tb~ of videos on my drive.
Thanks. My issue is that I'm on a Mac and so if this is PngOptimizer then I wouldn't be able to run it anyway. Unless it's compatible with Wine etc. But, yes, if you're looking for lossless compression that would be the way to go.