  An $n$-crossing projection of a link $L$ is a projection of $L$ onto a plane
such that $n$ points on $L$ are superimposed on top of each other at every
crossing. We prove that for all $k \in \mathbb{N}$ and all links $L$, the
inequality $$c_{2k+1}(L) \geq \frac{2g(L) + r(L)-1}{k^2}$$ holds, where
$c_{2k+1}(L)$, $g(L)$, and $r(L)$ are the $(2k+1)$-crossing number, $3$-genus,
and number of components of $L$ respectively. This result is used to prove a
new bound on the odd crossing numbers of torus knots and generalizes a result
of Jablonowski.
  We also prove a new upper bound on the $5$-crossing numbers of the 2-torus
knots and links. Furthermore, we improve the lower bounds on the $5$-crossing
numbers of $79$ knots with $2$-crossing number $ \leq 12$. Finally, we improve
the lower bounds on the $7$-crossing numbers of $5$ knots with $2$-crossing
number $\leq 12$.
