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Suppose $G$ is a non planar graph such that $G/e$ is planar for every edge. then want to show that at most six vertices of $G$ have degree 3 or more.

Well I know that a G is non planar iff it contains a subdivision of $K_{5}$ or $K_{3,3}$ as a subgraph, also G itself must be this subgraph because if not it would contradict that $G/e$ is planar.

So is that enough of to make the conclusion?

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    If G was not the subgraph that contained a subdivision, but removing an edge made it planar, would be a contradiciton – 2017-02-14
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    you showed that $G$ is $K_{3,3}$ or $K_{5}$. That is not exactly true, since $G$ may also contain isolated nodes. This is sufficient to prove your thesis.. – 2017-02-14
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    @Exodd Please explain what is meant by it is sufficient but also not true – 2017-02-14

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Following your reasoning, $G$ must contain a subgraph $G'$ that is (a subdivision of) $K_{3,3}$ or $K_5$. If $G$ also contains an edge $e$ not in $G'$, then $G/e$ is not planar, that is an absurd.

This means that the edges of $G$ concide with the edges of $G'$, but the same is not true for the nodes. In fact $G$ may contain nodes of degree 0 that are not included in $G'$.

In any case, $G$ contains only 5 or 6 nodes with degree greater or equal than 3, that are the nodes in $G'$ (without conunting the ones in the subdivision that have degree 2), so the thesis is true.