I don't have thick or curly hair, so I don't have any personal experience to share. However, back in my younger days, my hair was much thicker than it is now and I do remember some of the issues. (Assuming I've not become totally senile in the meantime!
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My single braid back then was quite heavy and actually pulled my head back at times. To remedy this, I would do 2 braids, starting just behind my ears, or do twin french braids.
When I needed my hair "out of my face" and I didn't want braid whips to contend with, I would pin the twin braids around my head. If your braids are not long enough to wrap the head entirely, then try to wrap just half the head with one braid, then the other half with the other braid.
French braids look fantastic with thicker, curly hair (and it usually stays in the braid better than straight, slick hair).
You could also try an inverted french braid. You have to be a bit of a contortionist to do this, but the results are really nice. Start with your head down and your hair combed from the nape of your neck up over your head (think upside down). Start french braiding your hair from the nape, working your way to the crown, or top of your head. Depending on how long your hair is, you could tuck the ends down, inside the french braid, or use a barrette to hold the finished braid to your head and let the curly ends "do their thing"!
Good luck and if you come up with anything that works for you, let the rest of the thick & curly world know about it!