Can a planar graph be drawn with all vertices on a straight line?

I have been repeatedly trying to prove and disprove the following:

Can any planar graph, with $n$ vertices, be drawn such that the vertices are fixed at coordinates $(0,0)$, $(1,0)$, ..., $(n-1,0)$ and all edges are half circles. (or convex/concave curves under/over the $y=0$ line).

Another way to express the problem, is whether we can always draw a planar graph, such that all vertices are on a straight line, and the edges are fairly simple (no zig-zag). There is (of course) no ordering on the vertices, so any permutation will do.

I can draw $K_4$, any subdivision and any planar graph I have tried, this way. However I haven't been able to find a proof.

I have tried many kinds of induction, e.g. triangulation based, but no luck. Do you have any ideas? (Or can you see a counter example?)

Update: As Nick wrote in the answer, the correct nomination appears to be "Book graph" and the following is a counterexample to the statement that all planar graphs are "2-bookable". Thanks Nick!

I'm pretty sure the condition you're imposing is equivalent to looking for a book embedding with book thickness at most 2. As proven in this paper, this is the case if and only if your graph is a subgraph of a Hamiltonian planar graph. The Goldner-Harry graph gives an example of a non-Hamiltonian, maximal planar graph. I do believe it is an example where your statement is false.