Have been training Muay Thai for about 4 months, yet I still find myself struggling with almost all the Muay Thai kicks - especially the low and body kick. My kicks happen to be weak as I often lose balance while trying to pivot my foot while kicking.
What are some ways I can focus exclusively on improving my Muay Thai kick technique?
The first and most important part is: Ask your instructor. I know that there are plenty of martial arts instructors out there that are not as able as they should be in helping their students and addressing the particular problems and needs they have. This is actually quite common. But obviously, we cannot possibly give personalised advice here, not only because this has to happen on a personal basis, but also because StackExchange forbids that as off-topic.
That being said: You speak of losing your balance. Losing the balance while kicking (or throwing) is very common among people that are in their first steps of experiencing martial arts. Actually, the later they start without former training in martial arts or e.g. gymnastics, the worse these problems are.
The reason for this problem is indeed one you can work on without much risk by yourself, at home: your muscular frame is unstable. This will not change fast, it will take time. And you can shorten this time by doing the exercises suggested below.
This stability, built from the feet upwards, relies on small, diagonal muscle fibres (as opposed to the "big players" like the quadriceps). To train them and gain stability, there are some simple exercises you can do which will not take you more than quarter an hour, done 3 days a week (for regeneration/buildng up, thus ideally coinciding with your training days). These exercises have the advantage that they a) can be done without much risk of harming you (IF you follow the steps one by one, do not proceed until you can do three sets over 90 seconds each!) and b) will not interfere with the dynamic movement patterns you aim to learn in your martial arts training and which should only be trained under the guidance of a correcting coach, they are supplementary:
Standing on one leg: All these things are done as long as you can or, if you make it, up to 90 seconds, with three sets per leg. The first step is being able to stand on one leg for extensive amounts of time (more than one minute) at all. Untrained persons will feel the muscles in their lower leg quite fast. If that does not pose a problem anymore, additionally move your lifted foot slowly from as far to the front as you can to as far to the back as you can. Next step would be slowly drawing "eights" (circle in front, transitioning to circle in back) in both directions, changing the direction after about half the time. If you can make three sets à 90 seconds per leg with this, you could add one-legged squatting and start the steps all over again. But you should already be much more stable at that time.
Static "kicking" (not before stable standing on one leg!): turn your foot before lifting the leg to the point where it should be turned during a kick, lift and hold the leg to the side as far up as you can. If you can do this without problems through all three sets, you can start to pivot your foot from normal to kicking direction and back while holding your foot still up. Supplementarily, you can use a rubber band (low resistance) between your legs if the lifting and holding as such do not pose a problem to you anymore as a second step, but this is not essential. As with the first exercise, up to 90 seconds per leg, three sets each.
Doing this on a regular basis, you should feel a betterment in your stability within a few months. I know practitioners that have these problems after years because either their coaches do not even realise the causing problem (muscular instability) or the practitioners do not have the time or will to work on that. Given stability, you will see that your head and body can concentrate on coordinating the muscles for more powerful kicks much easier. Without a stable basis, your body will be occupied with regaining balance more than with the actual kick.
Kicks of this kind can usually be performed by setting the foot first. So you step towards the opponent, placing one foot at a pre-twisted angle and then raise the other knee and twist the hip to kick round.
I'm not saying this is the best way to kick but it's certainly helpful for beginners to learn to balance.
Another tip is to keep whatever leg you are standing on bent slightly. It make balancing while kicking a whole lot easier.
Practice lifting your knee to your chest twisting your bottom half of your body to the side and kicking up while keeping your fist and arms balanced to your front.
