Show there exists a family of graphs, universal constants $c$ and $c'$ s.t. for each graph with at most $cn$ edges, $n$ being the number of vertices, there are at least $c'n^2$ crossings.
I have tried to use complete graphs as complete graphs have ~$cn^4$ (n choose 4) crossings and ~$n^2$ (n choose 2) edges but this does not satisfy the condition of each graph having at most $cn$ edges. Complete bipartite graphs do not work for the same reason. I am stuck to think of which kind of graph has this property.
Take a graph consisting of a complete graph $K_t$ and $n-t$ isolated vertices, for suitably chosen $t$.
Note that complete graphs actually have crossing number less than $\binom n4$, but the crossing number inequality still gives an $\Omega(n^4)$ lower bound for them, so this shouldn't affect you except in the choice of $c$.