  We develop new tools to analyze the complexity of the conjugacy equivalence
relation $E_\mathsf{lo}(G)$, whenever $G$ is a left-orderable group. Our
methods are used to demonstrate non-smoothness of $E_\mathsf{lo}(G)$ for
certain groups $G$ of dynamical origin, such as certain amalgams constructed
from Thompson's group $F$. We also initiate a systematic analysis of
$E_\mathsf{lo}(\pi_1(M))$, where $M$ is a $3$-manifold. We prove that if $M$ is
not prime, then $E_\mathsf{lo}(\pi_1(M))$ is a universal countable Borel
equivalence relation, and show that in certain cases the complexity of
$E_\mathsf{lo}(\pi_1(M))$ is bounded below by the complexity of the conjugacy
equivalence relation arising from the fundamental group of each of the JSJ
pieces of $M$. We also prove that if $M$ is the complement of a nontrivial knot
in $S^3$ then $E_\mathsf{lo}(\pi_1(M))$ is not smooth, and show how determining
smoothness of $E_\mathsf{lo}(\pi_1(M))$ for all knot manifolds $M$ is related
to the L-space conjecture.
