I practice trills every day. You should be practicing the 13231323 type trills, as long term it offers relaxation built into the fingering as 1-3 don't share a tendon. It's slower at first but with a little practice it'll feel like a breeze. To practice trills I do two things. First is to do repeated notes exercises with a single finger (not holding down other notes while I do them). To me a trill is basically repeated notes on alternating fingers--that's basically your speed limit. My first step in practicing trills is to do two quick pulses on a note with a single finger, then three, then four, then three, then two, the cycle twice for every finger. Then I go into trill exercises that have three components. First is a warmup where I slur one note into the adjacent note (the note I would trill it with), like B into C--then I do the reverse. These are like grace note appoggiature and should be able to be sounded pretty much instantaneously once you get used to them. Then I practice turns (3 note trill, like B,C,B, pause then C,B,C pause, repeat). Finally I do 5 note trills (B,C,B,C,B pause C,B,C,B,C, repeat). When I do this trill practice, I first do it with adjacent fingers (1,2; 2,3; 3,4; 4,5) and then do it with every other finger (1,3; 2,4; 3,5). The latter are optimal trill fingerings because they don't share tendons so allow for less tension. Further down the line you can link up the practiced trills to get longer ones, but I find a 5 note trill is a good start to get the mechanics down.
I've been doing this for a few months, not for basic 2-3 or 1-3-2-3 trills but for 3-4, 4-5, and 3-5 that come up in repertoire that I've struggled with. I've seen a marked improvement with those weaker-fingered trills and more relaxation with more standard trills. It's opened up repertoire for me and given me confidence. Trills, like a lot of technique, are a work in progress that take time to be really fluent with. It's probably less important what exercises you do to improve them than it is simply that you're taking time to work on your trill, provided that you're not doing something laden with tension.