Is it possible to avoid the Runge phenomenon by other norm instead of $L_{\infty}$ norm?

Question

Most of error analysis of polynomial interpolation employs $L_{\infty}$ norm. However, $||\prod\limits_{i=1}^{n}(\cdot-x_i)||_{L^{\infty}(a,b)}\leq\frac{n!h^{n+1}}{4} $ when equidistant points are used, i.e., $x_i=a+ih$ with $h=(b-a)/n$. The bound $\frac{n!h^{n+1}}{4}$ is not very small compared that in Chebyshev nodes. My question is that is it possible to reduce the bound for equidistant points when applying $||\prod\limits_{i=1}^{n}(\cdot-x_i)||_{L^{p}}$ for some $p<\infty$?