Hi nucci60,
Yes, it's normal for the branches to bend from the weight of the flowers. As you can see from the picture of one of mine, they can get weighed down quite a bit. In case you notice the two flower colors, they were seedlings when I transplanted them together. I didn't know at the time they were different colors but they look nice together.
BTG

If you don't like the look of the drooping, you can prune the plant down early next spring in March.
The general pruning rule is 1/3.
Rose of Sharon blooms on new wood only, so some people cut it way back, to half or almost a foot above the ground.
I know a neighbor who cut her ROS to about 18 inches high.
It grew alot that year, and really looks nice.
That way, it flowers from all the new growth and won't bend over like that.
If you are afraid of pruning it, just prune off the top third of the bush, before it leafs out around Saint Patty's day.(that is when I prune everything, I can remember better).
It grows in real nice, and I think you will be pleased at how it looks.
Then the following year, prune it again by a third.
Rose of Sharon is very hardy, so don't be afraid to prune it.
Good Luck!

Check out the link below for pruning instructions,
someone else on this forum posted this a long time ago, it is very helpful with all of your shrubs.

Here is a link that might be useful: Pruning Techniques

I let a RoS seedling grow several years ago. It gets a drooping, shaggy look even before buds && blooms start. Thought it was a result of some power washing done but now I think its a mutation. I may have the start of a true weeping Rose of Sharon. What do you think? Perhaps i should try rooting some cuttings & see if it continues.

and from a few years ago