  We prove that the r-th pluricanonical maps of threefolds of general type have
birationally bounded fibers if $r\geqslant 2$. Similarly, we prove that the
r-th pluricanonical maps of fourfolds of general type have birationally bounded
fibers if $r\geqslant 4$. We extend these results to higher dimensions in terms
of constants arising naturally from the birational geometry of varieties of
general type.
