Thank you for your question about your rose of Sharon shrub.  from your photo is looks like your tree branches are covered with lichens.  Lichens are a combination fungus/algae organism that does not harm branches that they inhabit.  While they do not harm the plants they inhabit, they can indicate that the plant is in decline (for other reasons).  Here's a link to an article about lichens from the University of Maryland Extension:
https://extension.umd.edu/resource/lichen-algae-and-moss-trees/

OSU tells us that "with advanced age, trunks may develop cankers that cause individual branches to die, followed by decline and death of the entire plant".  I suspect that this is the cause of your branch die-back.  

Here is the publication in full---it has a lot of good information about growing rose of Sharon:
https://plantfacts.osu.edu/tmi/Plantlist/hi_iacus.html

Here's another publication that has a good list of issues that affect rose of Sharon:
https://www.gardeningknowhow.com/ornamental/shrubs/rose-of-sharon/rose-of-sharon-problems.htm

Good luck with your rose of Sharon.