  We provide the first nontrivial upper bound for the chemical distance
exponent in two-dimensional critical percolation. Specifically, we prove that
the expected length of the shortest horizontal crossing path of a box of side
length $n$ in critical percolation on $\mathbb{Z}^2$ is bounded by
$Cn^{2-\delta}\pi_3(n)$, for some $\delta>0$, where $\pi_3(n)$ is the
"three-arm probability to distance $n$." This implies that the ratio of this
length to the length of the lowest crossing is bounded by an inverse power of
$n$ with high probability. In the case of site percolation on the triangular
lattice, we obtain a strict upper bound for the exponent of $4/3$.
  The proof builds on the strategy developed in our previous paper, but with a
new iterative scheme, and a new large deviation inequality for events in annuli
conditional on arm events, which may be of independent interest.
