We have, by one of Turán's theorem, that among the $n$-vertex simple graphs with no $r+1$ clique, the complete $r$-partite graph which has its number of vertices in each partite sets differing by at most $1$ vertex(The Turán graph) is the one that has maximum number of edges.
Now, by Kuratowski's theorem, a planar graph should not contain a $5$ clique or its topological equivalents. Why cant we apply the Turán's theorem to prove that planar graphs can be partitioned into at the most four independent vertices? Specifically, using $r=4$ would help us do the job, I think? The case of graphs like Mycielski graphs could be handled by showing them topologically equivalent to $K_{3,3}$, which arent allowed by Kuratowski theorem. Any clarifications as where this approach may fail? Thanks beforehand.