Is there a way of making normal paints more like a glaze? I've heard some people do this by thinning them down (either with water or medium), but of course that's the normal way to make a wash.
I have tried that (with water) only to get it running off, and/or leaving tide marks. I've not managed the glaze/tint that I was after, but maybe I'm doing it wrong.
It sounds like you're maybe applying your glaze too heavily.
Though a wash/glaze will be of a similar consistency, how you a apply it to the mini - and what it does when it's on there - is actually quite different. Where a wash is designed to achieve an intentionally uneven finish (by running into the recesses), you want a glaze to be as flat and even a layer as possible. Where you can often load your brush up with wash, slap it on the model, and let it do its thing, with a glaze you want to be wicking most of it off your brush before you apply it to the model so that it stays where you put it - still wet, and smooth, just very controlled. This is where the choice of medium can aid you. A Citadel shade, for example, is well suited for glazing as it's thin and transparent. You have to be careful, however, because its medium is designed to have a low surface tension (lower than water) to help run into the recesses. Glaze medium, on the other hand, does not have this problem and also slows the drying time of the paint so that it stays wetter on the brush, even after you've wiped most of it off, which is particularly important if you're doing a larger area.
To answer your question, you can absolutely make your ordinary paints into a glaze consistency, even just with water. The more transparent your starting colour, the easier a time you'll have; white, or anything with white in it, is lot harder to work with because it's so opaque. Reds and purples usually work well. This is why inks are a good starting point; aside from white most colours are very transparent.
I learned a lot about glazes (and a lot of other things) from
Vince Venturella. He has a couple of videos on the subject, if you've got 45 mins:
Understanding paint transparency, glazing and underpainting was one of those things that seemed super obvious once I'd seen a few videos about it, and has opened up a lot of techniques for me.