Abstract: In this paper, we investigate the existence and multiplicity of normalized solutions for the following fractional Schrödinger equations $$\begin{equation*} \begin{cases} (-\Delta)^{s} u+\lambda u=|u|^{p-2}u-|u|^{q-2}u,\ \ x\in \mathbb{R}^{N},\\ \displaystyle \int_{\mathbb{R}^{N}}|u|^{2}{\rm d}x=c>0,\\ \end{cases}\tag{$P$} \end{equation*}$$ where $N\geq 2$, $s\in (0,1)$, $2+\frac{4s}{N}<p<q\leq 2_{s}^{*}=\frac{2N}{N-2s}$, $(-\Delta)^{s}$ represents the fractional Laplacian operator of order $s$, and the frequency $\lambda\in \mathbb{R}$ is unknown and appears as a Lagrange multiplier. Specifically, we show that there exists a $\hat{c}>0$ such that if $c>\hat{c}$, then the problem ($P$) has at least two normalized solutions, including a normalized ground state solution and a mountain pass type solution. We mainly extend the results in [Commun Pure Appl Anal, 2022, 21: 4113-4145], which dealt with the problem ($P$) for the case $2<p<q<2+\frac{4s}{N}$.

Key words: fractional Schrödinger equation, normalized solutions, variational methods, competing power