  In this paper, we establish upper bounds on the length of the shortest
conjugator between pairs of infinite order elements in a wide class of groups.
We obtain a general result which applies to all hierarchically hyperbolic
groups, a class which includes mapping class groups, right-angled Artin groups,
Burger--Mozes-type groups, most $3$--manifold groups, and many others. In this
setting we establish a linear bound on the length of the shortest conjugator
for any pair of conjugate Morse elements. For a subclass of these groups,
including, in particular, all virtually compact special groups, we prove a
sharper result by obtaining a linear bound on the length of the shortest
conjugator between a suitable power of any pair of conjugate infinite order
elements.
