  We derive lower bounds for the $L^p(\mu)$ norms of monic extremal polynomials
with respect to compactly supported probability measures $\mu$. We obtain a
sharp universal lower bound for all $0<p<\infty$ and all measures in the
Szeg\H{o} class and an improved lower bound on $L^2(\mu)$ norm for several
classes of orthogonal polynomials including Jacobi polynomials, isospectral
torus of a finite gap set and orthogonal polynomials with respect to the
equilibrium measure of an arbitrary non-polar compact subset of $\mathbb{R}$.
