  We study the complexity of birational self-maps of a projective threefold $X$
by looking at the birational type of surfaces contracted. These surfaces are
birational to the product of the projective line with a smooth projective
curve. We prove that the genus of the curves occuring is unbounded if and only
if $X$ is birational to a conic bundle or a fibration into cubic surfaces.
Similarly, we prove that the gonality of the curves is unbounded if and only if
$X$ is birational to a conic bundle.
