  In the present paper, we study the following Schr\"{o}dinger-Maxwell equation
with combined nonlinearities
  \begin{align*}
  \displaystyle - \Delta u+\lambda u+ \left(|x|^{-1}\ast |u|^2\right)u
  =|u|^{p-2}u +\mu|u|^{q-2}u\quad \text{in} \
  \mathbb{ R}^3 \quad \quad \text{and}\quad \quad
  \int_{\mathbb{R}^3}|u|^2dx=a^2,
  \end{align*} where $a>0$, $\mu\in \mathbb{R}$, $2<q\leq \frac{10}{3}\leq p<6$
with $q\neq p$, $\ast$ denotes the convolution and $\lambda\in \mathbb{R}$
appears as a Lagrange multiplier. Under some mild assumptions on $a$ and $\mu$,
we prove some existence, nonexistence and multiplicity of normalized solution
to the above equation. Moreover, the asymptotic behavior of normalized
solutions is verified as $\mu\rightarrow 0$ and $q\rightarrow \frac{10}{3}$,
and the stability/instability of the corresponding standing waves to the
related time-dependent problem is also discussed.
