Boundedness of fibers for pluricanonical maps of varieties of general type

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