  We show that if a graph $G$ with $n \geq 3$ vertices can be drawn in the
plane such that each of its edges is involved in at most four crossings, then
$G$ has at most $6n-12$ edges. This settles a conjecture of Pach,
Radoi\v{c}i\'{c}, Tardos, and T\'oth, and yields a better bound for the famous
Crossing Lemma: The crossing number, $\mbox{cr}(G)$, of a (not too sparse)
graph $G$ with $n$ vertices and $m$ edges is at least $c\frac{m^3}{n^2}$, where
$c > 1/29$. This bound is known to be tight, apart from the constant $c$ for
which the previous best lower bound was $1/31.1$. As another corollary we
obtain some progress on the Albertson conjecture: Albertson conjectured that if
the chromatic number of a graph $G$ is $r$, then $\mbox{cr}(G) \geq
\mbox{cr}(K_r)$. This was verified by Albertson, Cranston, and Fox for $r \leq
12$, and for $r \leq 16$ by Bar\'at and T\'oth. Our results imply that
Albertson conjecture holds for $r \leq 18$.
