Q.why is my burning bush dying?
The burning bush is a mature (about 16 years) plant. Leaves have developed darkish blotches which gradually grow bigger. The leaves eventually die and then the whole branch dies. It can be snapped off.
This started at one side of the plant but is spreading across it.
What is causing the problem?
What can I do about it?

Certified GKH Gardening Expert
It could be a fungal spot disease. Try spraying with a fungicide and see if that helps. Prune affected leaves and branches as you are doing. Avoid overhead watering.
You might also inspect the plant for scale, which are common pests of euonymus. If found, treat with neem oil.
Another possibility that causes branch dieback is verticillium wilt. You could contact your local extension agent for confirmation. About the only thing you can do is cut off the diseased branches to healthy wood.
Root rot can also cause branch dieback.
https://www.gardeningknowhow.com/ornamental/shrubs/burning-bush/burning-bush-care.htm
https://www.gardeningknowhow.com/ornamental/shrubs/burning-bush/burning-bush-turning-brown.htm