Let $f, u$ be non-constant rational functions over algebraically closed field of characteristic 0 with $\deg{f}$ at least 2. We consider reducibility of the sequence of curves $C_n : f^n(x) = u(y)$. Our main result is that there exists a constant $b$ depending only on $\deg(u)$ such that if $C_m$ is irreducible for some $m \geq b$, then $C_n$ is irreducible for all $n\geq 0$.