k-outerplanar graphs are subgraphs of bounded diameter planar graphs? of bounded diameter bounded genus graphs? 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.
No: see the nested triangles graph. It has bounded treewidth (more strongly, bounded pathwidth and bandwidth) but cannot be augmented to have bounded diameter while preserving planarity.