It is well known that bounded diameter planar graphs have bounded treewidth (e.g. see this). Does the converse hold? That is, is every planar bounded treewidth graph the (induced?) subgraph of a bounded diameter planar graph?
I am interested in a proof sketch/reference if this is true or a counterexample if this is false.
Note1: David's correct answer below is in the negative. In its light can we modify the question to read: Are k-outerplanar graphs subgraphs of simultaneously bounded diameter and bounded genus graphs?
Note2: David further indicates (correctly) that the nested triangles graph cannot be made bounded tree-width even at the expense of increasing the genus to a constant.