C.G.
The exact same thing happened to my son when he was that age. he also has a full head of hair, and it happened again when he was around three. Both times I washed his hair with a mild Head and Shoulders, and bought a very soft brush, I put a tiny drop of tea tree oil in his scalp where the dry patched were and brushed it through with the soft brush. Do it at night on dry hair and I promise the hair will not be be oily in the morning, because it will be absorbed into the scalp. Took about a week and and his scalp was fine. To prevent it now, I wash his hair about once a week with the medicated shampoo (he's 5 now)