This answer, as pointed out by Jan in the comment, is incorrect, as the definitions slightly differ. I leave it here as it contains useful pointers.
If I understood your definitions correctly, your $\mathcal{C}$ would be called an outerstring graph, and your $\mathcal{C}'$ a double outerstring graph. There are outerstring graphs that are not double outerstring graphs, in fact, even the order of the growth rates of these two families are different. For some literature and a nice summary of results, see the first two pages here: Alexandre Rok and Bartosz Walczak: Outerstring graphs are χ-bounded.