$L^p$-convergence when composed with a smooth function

Question

Let $L\in C^\infty(\mathbb R)$ be a smooth function, such that $L\circ f\in L^p(\mathbb R)$ for any $f\in L^p(\mathbb R)$. Now if $f_n\to f$ in $L^p(\mathbb R)$, does it implies that $L\circ f_n\to L\circ f$ in $L^p$?