Bucket circles, as well as circles in general, can be rather complicated initially to get a handle on. The majority of issues with circle training arise as to the fact that when performing circles you will essentially be combining a support with a swing; two elements that on first appraisal appear to be diametrically opposed. Circles are however well worth the effort; consistent training of this element will deliver enormous benefits to the practitioner in terms of shoulder girdle and core strength.
Here are a few tips to help your own circles and bucket work become more fluid:
1) Perhaps the most difficult aspect of circles for most people to grasp is that this element is a swing, NOT a support. The body will be "hanging" in between the hands. It is this "hang" that allows the body to swing fluid circles; much like a soft towel can be "snapped" - but a stiff bar cannot.
2) For this very reason, I introduce all of my athletes to circle training by having them do upper arm circles on the parallel bars. This allows them to immediately and clearly feel what I mean by the body hanging while performing circles.
To perform upper arm circles, begin from a straight arm support on the parallel bars. Now dip down until your shoulders are completely on top of your hands and then fall back until you are resting in a upper-arm support with your upper arms firmly on top of the rails. While in upper arm support, your hands will continue to hang unto the bars. In order to not fall through the bars, you will need to maintain firm pressure with the upper arms and shoulder girdle.
Now simply begin to swing your legs around in a circle. Please note that I said swing and not lift. There should be no segmentation of the body; the body should swing freely and smoothly as a single unit.
3) Once you feel that you have mastered upper arm circles enough to move onto bucket circles, I recommend that you lower the bucket until it is only a few inches off of the ground.
4) Strive to keep the arms firm, but the shoulder girdle relaxed. To feel this relaxed shoulder girdle, assume a straight arm support (the beginning position of a dip) on a dipping bar. Now keeping the arms straight, simply allow your shoulders to relax and your neck to sink downward in between the shoulders. There is no need to over exaggerate the degree of sag, just allow the body to do whatever is most natural and comfortable.
5) At your level of development, my recommendation is to not consciously push down during your circles; but to rather concentrate your focus on letting your body continue to "hang" while swinging. For beginners, pushing down will almost always cause you to either lift the feet in front, or lift the hips/back/shoulder girdle in the rear. There is a subtle, but substantial difference between leaning on the arms and pushing down with the arms.
6) The 3/4 of the circle is usually the most difficult part to perform correctly. The temptation to push down and lift the hips here to complete the circle is extreme; do not give in to it.
7) To make your circle larger and more fluid, either lean more to the sides during the circles, swing faster, or both.
8 ) A fun way to very quickly increase your proficiency in the bucket is to play "Add On" with a friend. To play Add On, the first player will perform a skill in the bucket. The next player will perform that skill plus then add a new one of his own. The game goes on in this back and forth manner, creating a longer and longer series of skills strung together. When someone makes an error (i.e. unable to complete a skill in the series), they get one additional attempt. If they miss again, they are out.
Pommel horse routines on YouTube are an excellent resource for seeing new skills to try in the bucket and to use in Add On. What would be impossible to perform without years of practice, now becomes plausible with the bucket and a great way to help you begin to break down the intricacies of pommel horse on your own. In addition, bucket training is just plain fun. My own athletes will occasionally play Add On in the bucket for an hour or more until they quite literally can no longer hold themselves up - and they will literally be laughing and smiling the entire time 8).
9) Extensive work in the bucket will require that you use the correct equipment. By far the best training mushrooms on the market today are the EGE versions. I believe in them so strongly that I have two of them in my gym. The EGE pommel trainers have a far greater curve to the top surface than traditional mushrooms which greatly diminishes the stress on the wrist from training. They come in two sizes; the red - (which is smaller and was originally designed as a special request and built to my specifications) and the black (larger and capable of holding two pommels)

The following
Yours in Fitness,
Coach Sommer