Let $G$ be a nonplanar graph (undirected) of $n$ nodes and $e$ edges, with
$e \le 3n-6$.
Define a *rewiring* move as replacing edge $(a,b)$ with edge $(a,c)$.
The result must be a simple graph (no loops, no multiple edges).
So after each rewiring, the number of vertices and edges is the same: $n$ and $e$.

I would like to find the fewest number of rewirings that converts $G$
to a planar graph.
An example is shown below.

^{ (a) $e=11 \le 3n-6=12$. (b) Edge $(1,3)$ rewired to $(1,6)$. (c) Planar embedding. }

My main question is: Has this notion been studied before, and if so, under what terminology? If it hasn't been studied, I would be interested in a proof that every such nonplanar graph can be rendered planar via rewirings, or a counterexample.

**Added**. Here is another example, of what Gerhard calls a balloon graph.
The five rewirings (b)
$$(1,3), \; (1,4), \; (1,5) \to (1,7), \; (1,8), \; (1,9)\\
(5,2), \; (5,3) \to (5,7), \; (5,8)
$$
obviously reach planarity, but just the three rewirings (c)
$$(1,3), \; (1,4) \to (1,7), \; (1,8) \\
(5,3) \to (5,7)
$$
less obviously result in a planar graph.
I don't think two rewirings suffice.

^{ (a) Nonplanar $G$. (b) $5$ rewirings: planar. (c) $3$ rewirings: planar. }