  We study the 1-planar, quasi-planar, and fan-planar crossing number in
comparison to the (unrestricted) crossing number of graphs. We prove that there
are $n$-vertex 1-planar (quasi-planar, fan-planar) graphs such that any
1-planar (quasi-planar, fan-planar) drawing has $\Omega(n)$ crossings, while
$O(1)$ crossings suffice in a crossing-minimal drawing without restrictions on
local edge crossing patterns.
