I am kinda new to graph theory so I appreciate any suggestion or hint to approach this question. Thank you!
Suppose a graph G does not have K2,2 as a subgraph. Suppose also that G has exactly 4 vertices of degree 4 , and all other vertices have degree less than 4. Explain why G must be planar.
It is possible for such a graph to be non-planar. Take $K_{3,3}$, add two edges, one to each partite set, then subdivide all edges. The resulting graph is non-planar (it has $K_{3,3}$ as a topological minor), has the degree sequence desired, and has girth 6, so it has no $K_{2,2} = C_4$ subgraph.