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 \(p \geq 2\), 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 \(2\leq p \leq m^{o(1)}\) 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.