  We present a deterministic $n^{2+o(1)}$-time algorithm that approximates the
crossing number of any graph $G$ of order $n$ up to an additive error of
$o(n^4)$. We also provide a randomized polynomial-time algorithm that
constructs a drawing of $G$ with $\text{cr}(G)+o(n^4)$ crossings. These results
are made interesting by the well known fact that every dense $n$-vertex graph
has crossing number $\Theta(n^4)$. Our work builds on a technique developed by
Fox, Pach and S\'uk, who obtained very similar results for the rectilinear
crossing number.
  The results by the aforementioned authors and in this paper imply that the
(normalized) crossing and rectilinear crossing numbers are estimable
parameters. Motivated by this, we introduce two graphon parameters, the
crossing density and the rectilinear crossing density, and then we prove that,
in a precise sense, these are the correct continuous analogs of the crossing
and rectilinear crossing numbers of graphs.
