  We present faster high-accuracy algorithms for computing $\ell_p$-norm
minimizing flows. On a graph with $m$ edges, our algorithm can compute a
$(1+1/\text{poly}(m))$-approximate unweighted $\ell_p$-norm minimizing flow
with $pm^{1+\frac{1}{p-1}+o(1)}$ operations, for any $p \ge 2,$ giving the best
bound for all $p\gtrsim 5.24.$ Combined with the algorithm from the work of
Adil et al. (SODA '19), we can now compute such flows for any $2\le p\le
m^{o(1)}$ in time at most $O(m^{1.24}).$ In comparison, the previous best
running time was $\Omega(m^{1.33})$ for large constant $p.$ For
$p\sim\delta^{-1}\log m,$ our algorithm computes a $(1+\delta)$-approximate
maximum flow on undirected graphs using $m^{1+o(1)}\delta^{-1}$ operations,
matching the current best bound, albeit only for unit-capacity graphs.
  We also give an algorithm for solving general $\ell_{p}$-norm regression
problems for large $p.$ Our algorithm makes
$pm^{\frac{1}{3}+o(1)}\log^2(1/\varepsilon)$ calls to a linear solver. This
gives the first high-accuracy algorithm for computing weighted $\ell_{p}$-norm
minimizing flows that runs in time $o(m^{1.5})$ for some $p=m^{\Omega(1)}.$ Our
key technical contribution is to show that smoothed $\ell_p$-norm problems
introduced by Adil et al., are interreducible for different values of $p.$ No
such reduction is known for standard $\ell_p$-norm problems.
