In this paper, we investigate the following Chern-Simons-Schr\”{o}dinger system \begin{equation*} -\Delta u+u+\left(\int_{|x|}^{\infty} \frac{h(s)}{s} u^{2}(s) d s+\frac{h^{2}(|x|)}{|x|^{2}}\right) u=f(u) \quad \text { in } \mathbb{R}^{2}, \end{equation*} where $h(s)=\int_{0}^{s} \frac{t}{2} u^{2}(t) dt$ and the nonlinearity $f\in C(\mathbb{R}, \mathbb{R})$ satisfies the Ambrosetti-Rabinowitz type condition. By using a combination of invariant sets method and the Ljusternik-Schnirelman type minimax method, we prove the existence of infinitely many sign-changing solutions. It is worth mentioning that the nonlinear term may not be 5-superlinear at infinity. In particular, it includes the power-type nonlinearity $|u|^{p-2}u$ with $p\in (4, \infty)$. This work also answers the open problem raised by Liu, Ouyang and Zhang (Nonlinearity \textbf{32} (2019), 3082-3111).