This paper is devoted to studying the existence of normalized solutions for the following quasilinear Schrödinger equation \(\begin{aligned} \begin{aligned} -\Delta u-u\Delta u^2 +\lambda u=|u|^{p-2}u \quad \textrm{in}\ \mathbb {R}^{N}, N=3,4 \end{aligned} \end{aligned}\) where \(\lambda \) appears as a Lagrange multiplier and \(p \in (4+\frac{4}{N},2\cdot 2^*]\) . The solutions correspond to critical points of the energy functional subject to the \(L^2\) -norm constraint \(\int _{\mathbb {R}^N}|u|^2\textrm{d}x=a^2>0\) . In the Sobolev critical case \(p=2\cdot 2^*\) , the energy functional has no critical point. As for \(L^2\) -supercritical case \(p \in (4+\frac{4}{N},2\cdot 2^*)\) : on the one hand, taking into account Pohozaev manifold and perturbation method, we obtain the existence of ground state normalized solutions for the non-radial case; on the other hand, we get the existence of infinitely many normalized solutions in \(H^1_r(\mathbb {R}^N)\) . Moreover, our results cover several relevant existing results. And in the end, we get the asymptotic properties of energy as a tends to \(+\infty \) and a tends to \(0^+\) .
