Summary: In this article, we are first concerned with the existence of multiple normalized solutions to the following fractional \(p\)-Laplace problem: \[ \begin{cases} (-\Delta)_p^s v + \mathcal{V}(\xi x) |v|^{p-2}v =\lambda|v|^{p-2}v+f(v)\quad \text{in } \mathbb{R}^N, \\ \qquad\qquad\qquad \displaystyle\int\limits_{\mathbb{R}^N} |v|^p \, dx =a^p, \end{cases} \] where \(a, \xi>0\), \(p \geq 2\), \(\lambda \in \mathbb{R}\) is an unknown parameter that appears as a Lagrange multiplier, \(\mathcal{V}: \mathbb{R}^N \to [0, \infty)\) is a continuous function, and \(f\) is a continuous function with \(L^p\)-subcritical growth. We prove that there exists the multiplicity of solutions by using the Lusternik-Schnirelmann category. Next, we show that the number of normalized solutions is at least the number of global minimum points of \(\mathcal{V}\), as \(\xi\) is small enough via Ekeland’s variational principle.