  We are concerned with the existence and asymptotic properties of solutions to
the following fourth-order Schr\"{o}dinger equation \begin{equation}\label{1}
{\Delta}^{2}u+\mu \Delta u-{\lambda}u={|u|}^{p-2}u, ~~~~x \in \R^{N}\\
\end{equation} under the normalized constraint
  $$\int_{{\mathbb{R}^N}} {{u}^2}=a^2,$$ where $N\!\geq\!2$, $a,\mu\!>\!0$,
$2+\frac{8}{N}\!<\!p\!<\! 4^{*}\!=\!\frac{2N}{(N-4)^{+}}$ and $\lambda\in\R$
appears as a Lagrange multiplier. Since the second-order dispersion term
affects the structure of the corresponding energy functional $$
E_{\mu}(u)=\frac{1}{2}{||\Delta u||}_2^2-\frac{\mu}{2}{||\nabla
u||}_2^2-\frac{1}{p}{||u||}_p^p $$ we could find at least two normalized
solutions to (\ref{1}) if $2\!+\!\frac{8}{N}\!<\! p\!<\!{ 4^{*} }$ and
$\mu^{p\gamma_p-2}a^{p-2}\!<\!C$ for some explicit constant
$C\!=\!C(N,p)\!>\!0$ and $\gamma_p\!=\!\frac{N(p\!-\!2)}{4p}$. Furthermore, we
give some asymptotic properties of the normalized solutions to (\ref{1}) as
$\mu\to0^+$ and $a\to0^+$, respectively. In conclusion, we mainly extend the
results in \cite{DBon,dbJB}, which deal with (\ref{1}), from $\mu\leq0$ to the
case of $\mu>0$, and also extend the results in \cite{TJLu,Nbal}, which deal
with (\ref{1}), from $L^2$-subcritical and $L^2$-critical setting to
$L^2$-supercritical setting.
