  We prove strong hybrid subconvex bounds simultaneously in the $q$ and $t$
aspects for $L$-functions of selfdual $\mathrm{GL}_3$ cusp forms twisted by
primitive Dirichlet characters. We additionally prove analogous hybrid
subconvex bounds for central values of certain $\mathrm{GL}_3 \times
\mathrm{GL}_2$ Rankin-Selberg $L$-functions. The subconvex bounds that we
obtain are strong in the sense that, modulo current knowledge on estimates for
the second moment of $\mathrm{GL}_3$ $L$-functions, they are the natural limit
of the first moment method pioneered by Li and by Blomer.
  The method of proof relies on an explicit $\mathrm{GL}_3 \times \mathrm{GL}_2
\leftrightsquigarrow \mathrm{GL}_4 \times \mathrm{GL}_1$ spectral reciprocity
formula, which relates a $\mathrm{GL}_2$ moment of $\mathrm{GL}_3 \times
\mathrm{GL}_2$ Rankin-Selberg $L$-functions to a $\mathrm{GL}_1$ moment of
$\mathrm{GL}_4 \times \mathrm{GL}_1$ Rankin-Selberg $L$-functions. A key
additional input is a Lindel\"of-on-average upper bound for the second moment
of Dirichlet $L$-functions restricted to a coset, which is of independent
interest.
