Photos make difficult diagnostic tools as they do not show the whole plant, but this appears to be a condition known as shothole.
OSU has a good description: Shothole has various causal agents, bacteria, fungus, and environmental stress. Often, the problem is limited to the lower branches or the shady side of the hedge/shrub. The reason is the leaves in those positions dry slowly.
The underlying cause of shothole is difficult to diagnose unless seen early on, before the spots turn brown, dry, and fall out. Many times, the cause is attributed to environmental stress. Thus, sprays are seldom applied.
A laurel does well in sun or shade, with good air circulation. It's best in moist, well-drained soil. An organic mulch at the base enhances root conditions.
You do not have to remove leaves or branches unless they are unsightly.
Feel free to contact our office if you have other questions.
Let me know if I can help you further!
Jefferson County Cooperative Extension Service
4200 Gardiner View Avenue
Louisville KY 40213