We got reduction graph to planar bounded treewidth graph, but this is unlikely to be true.
Let $H$, the planarizing gadget, be planar graph with four distinguished vertices $u,u',v,v'$ on the outer faces.
Take graph $G$ drawn on the plane. Add new vertex $S$, adjacent to all vertices of $G$. So far the diameter is at most two.
Replace each pair of crossing edges $(u,u'),(v,v')$ by new copy of the gadget $H$.
The resulting graph $G'$ is planar with diameter $D = 2\max(d(u,u'),d(v,v'))$ where $d$ is the distance in $H$.
The treewidth of $G'$ is $O(D)$, which is constant for fixed $H$.
Similar reduction with specially chosen $H$ is used to show NP-hardness of problems for planar graphs.
What is wrong with this reduction?
Correctness of the reduction is unlikely, because for bounded treewidth graphs a lot of graph invariants are computable in polynomial time and choosing suitable gadget $H$ might give relation between invariants of $G$ and $G'$, implying $P=NP$.
Added Example of 3-Coloring planarizing gadget is in this lecture p.1.
We cannot use it directly as gadget $H$ since the $S$ vertex increases the chromatic number, but there is potential approach to use it and then subdivide all edges $(S,t)$ twice, having the property that $S$ can be any color in a valid 3-coloring by adjusting the colors of the degree 2 vertices in the subdivision.