This was a great question!!! Now we have lots of helpful answers. I also have used fusible fleece for the sides of purses, it was thicker and not quite so stiff as several of those mentioned. For the bottom of the purses I've used x-ray plates, plastic canvas and now someone has given me one of those big plastic signs from a 'favorite' coffee shop. I'll just have to remember to cut it with a rounded edge so no points go through the fabric.
Again: thanks for asking the question. I've learned something new.