Crossing number$($cr$)$: The crossing number of a simple graph is the minimum number of crossings that can occur when this graph is drawn in the plane where no three arcs representing edges are permitted to cross at the same point. For an undirected simple graph $G$ with $n$ vertices and $e$ edges such that $e > 7n$ the crossing number is always at least $$cr(G)\geq \frac{e^3}{29n^2}$$
$(1)$ I was wondering is there exist any upper bound in general$?$ Here is a upper bound for the crossing number of the complete bipartite graph $K_{m,n}$. I need a proof of that result.$($if possible than provide a simple's one$)$
Thickness: The thickness of a simple graph G is the smallest number of planar subgraphs of G that have G as their union.
$(2)$ I was wondering is there any relation between these two number$?$
Very late to the party, but here we go. (i) Clearly, as the addition of edges does not decrease the number of crossings, the graph on $n$ nodes with the highest crossing number is $K_n$. The current upper bound can be found on Wikipedia. (ii) The cube connected cycles are degree three graphs (hence they have thickness two) but they have non-constant crossing number, hence there is no clear relation.