The two island names sound like a couple of exotic Pirate’s isles in the south Caribbean but these two islands were once located south of the Main Street Bridge in the Miami River in Moraine.
You can see both islands clearly in the vintage photo. Cozy Island was small and just off Miami Shores.
Wizard Island was larger and located close to the eastern shore just north of the mouth of Holes Creek. Two senior citizens that I spoke to back in 2002 told me when they were kids in the summer they would canoe or swim out to Wizard Island. They would play for long periods of time or fish from the western side of the island.
Sadly, around 1959, both small islands were dredged up and the dirt from them was used to form the protective levee. On the plus side, this section of the Miami River was now wider and safer for boating and for seaplanes to land.

