  Based on Nijenhuis-Richardson bracket and bidegree on the cohomology complex
for a Lie conformal algebra, we develop a twisting theory of Lie conformal
algebras. By using derived bracket constructions, we construct
$L_\infty$-algebras from (quasi-)twilled Lie conformal algebras. And we show
that the result of the twisting by a $\mathbb{C}[\partial]$-module homomorphism
on a (quasi-)twilled Lie conformal algebra is also a (quasi-)twilled Lie
conformal algebra if and only if the $\mathbb{C}[\partial]$-module homomorphism
is a Maurer-Cartan element of the $L_\infty$-algebra. In particular, we show
that relative Rota-Baxter type operators on Lie conformal algebras are
Maurer-Cartan elements. Besides, we propose a new algebraic structure, called
NS-Lie conformal algebras, that is closely related to twisted relative
Rota-Baxter operators and Nijenhuis operators on Lie conformal algebras. As an
application of twisting theory, we give the cohomology of twisted relative
Rota-Baxter operators and study their deformations.
