Answer: This is not a healthy plant, based on your description, normally it blooms and the foliage stays healthy until the fall. When a plant initially seems fine and then goes into stress in the heat of summer, usually we look for moisture stress such as caused by underwatering or by a root problem.
There is a chance it is not rooting into the surrounding soil, if for examnple it had encircling roots that were not cut and/or untangled and directed outward at planting. This would make it impossible to take up enough water once the really hot weather starts.
Another possibility is that although you are watering it is not deep and thorough. This plant likes an evenly moist soil but will tolerate average moisture if it has shade in the afternoon. In a hot sunny location you need to pay special attention to watering. To know if you need to water, dig into the soil with your finger. If it is still damp do not water yet. When you do water, water slowly so it soaks down deep. It is better to water deeply less often than to water daily. After watering, wait a few hours and then dig down to see how far the water went -- it can be surprising. Using a few inches of organic mulch over the root system can also help keep the soil moister longer.
This plant is usually considered to be quite pest and disease free, however it is also possible that something unusual is bothering it. If your watering has been adequate, you might want to check with your county extension and see if they can help you diagnose the cause of the browning. If it is something that requires a chemical control, they will have the most up to date information on what to use and how/when is best to use it. I'm sorry you are having trouble with your Clethra, it is such a great garden plant.
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