Why has my cedar shrub turned half brown?
You didn't say when you noticed the browning. If it occurred over winter, it is probably winter burn, caused by exposure to late winter sun, although salt damage is also possible. Winter burn is very common on arborvitae and cedar. You can prune out the dead part. Whether the plant will ever look good again will depend on how extensive the damage. In the future, protect evergreens from late winter sun with some kind of sunshield. Some even wrap their plants in burlap:
If it's not winter burn, check this link for diagnostic help, and write back with more details if this link doesn't help: