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Hello everyone,
this bonsai (ligustrum) belongs to my parents. 2 months ago it got ill (I do not have any pictures). After the description my parents gave me, it seemed to be powdery mildew. But I might be mistaken. Anyway, they sprayed it with a common insecticide, and after that the top branch dried. I noticed yesterday that another branch is drying. Might have been over watered. Is there anything I can do to try to save this bonsai?
Thank you all for reading.

There is no point in spraying with insecticide if it is mildew! I can't tell from the images what the problem might be, It could be anything, environment, insects. You should be able to see something on the leaves or trunk and be able to tell what it is,

what I can tell from your pictures it looks like woolly aphids are affecting your tree. Please check whether this is indeed what you have on your plant and treat accordingly.

Make sure you water sparingly, but avoid full drying out of the substrate.

I will try to take a better photo of the disease. My main concern are the dried branches. Should I cut them?

Extra magnified images.

According to my parents, there were small white jumping fleas which produced something like white cotton on the branches/leaves. After spraying insecticide, they died, but the tree's health declined and branches started to dry.

Do not cut.

What you got is called aphids.
Aphids effekts the tree by getting black leaves that falls off.
Remove the cotton on the branches with a wet cloth.
That should do it If you dont have fleas anymore.

cotton bud and methalated spirit will do it, but you need to do it every ten days.