  We prove quantitative sub-ballisticity for the self-avoiding walk on the
hexagonal lattice. Namely, we show that with high probability a self-avoiding
walk of length $n$ does not exit a ball of radius $O(n/\log{n})$. Previously,
only a non-quantitative $o(n)$ bound was known from the work of Duminil-Copin
and Hammond \cite{DCH13}. As an important ingredient of the proof we show that
at criticality the partition function of bridges of height $T$ decays
polynomially fast to $0$ as $T$ tends to infinity, which we believe to be of
independent interest.
