Certified GKH Gardening Expert
From my limited research, it would appear that you may have had a significant amount of rainfall in your area this year, or possibly at the end of last. This would tell me that it may be a fungal infection or blight of sorts.
Ligustrum are usually pretty hardy, and resistant to disease, but heat coupled with a wet period is guaranteed to produce an infection.
Fortunately, the cure will be relatively easy. Apply a HANDFUL of DOLOMITIC LIME and a TABLESPOON of WETTABLE SULFUR per shrub. Scatter this through the soil.
After, you will want to use a spray fungicide for the upper portions of the plant. This article will help:
This article will help you with the general care of Ligustrum: https://www.gardeningknowhow.com/ornamental/shrubs/ligustrum/growing-ligustrum-shrubs.htm