Yes, a Turkish type horn bow fits in complex composite, but if shot with a siper overdraw and thumb ring, then it can be shot as a "Turkish".
Self: bow limb made from single piece or billets spliced at the handle. Tip overlays less than 2" long are allowed.
Simple Composite: Simple backed bows, or sinew backed self bow fits here. Limb construction is no more than two laminations.
Complex Composite: Any combinations of natural materials. No metal or metal fasteners allowed however. Non-mechanical releases okay as long as made from natural materials and/or metal (no plastics). Metal is only allowed for the release device and arrow points.
Turkish: Must be shot with a thumb ring and siper overdraw.