This is the underside of one of the layouts I had made. I used 2 layers of 1/8" hardboard (hardboard tempered panel).
It is a denser version of MDF, but flexible. The 4" strips were glued to edging and pinned with brads and micro pins. Small plywood blocks were added for additional support, especially at joints. The second layer was glued to first. Butt joints were clamped, to make sure they stay and not lift. Try not to have joints in a tight radius location. The top edge was skimmed with wood filler and painted.
You can also get a product called wiggle wood or bending plywood. It works well but the face can be grainy and need a bit of prep. In my case the hardboard was cheaper.