Let $p(x)$ be a degree $n$ polynomial over $[-1, 1]$, and let $q(x) = p'(x) \sqrt{1-x^2}$. Is it true that $$ \|q\|_1 \leq O(n) \|p\|_1 $$ where we define $\|f\|_p := \left(\int_{-1}^1 |f(x)|^pdx\right)^{1/p}$?

For reference, Bernstein's inequality shows that $$ \|q\|_\infty \leq n\|p\|_\infty $$ with equality at the $n$th Chebyshev polynomial, and looking at Legendre polynomials shows that $$ \|q\|_2 \leq \sqrt{n(n+1)}\|p\|_2 $$ with equality at the $n$th Legendre polynomial.