  We study the existence and nonexistence of normalized solutions $(u_a,
\lambda_a)\in H^{1}(\mathbb{R}^N)\times \mathbb{R}$ to the nonlinear
Schr\"{o}dinger equation with mixed nonlocal nonlinearities.
  This study can be viewed as a counterpart of the Brezis-Nirenberg problem in
the context of normalized solutions to the nonlocal Schr\"{o}diger equation
with a fixed $L^2$-norm $\|u\|_2=a>0$.
  The leading term is $L^2$-supercritical, that is, $p\in
(\frac{N+\alpha+2}{N},\frac{N+\alpha}{N-2}]$, where the
Hardy-Littlewood-Sobolev critical exponent $p=\frac{N+\alpha}{N-2}$ appears.
  We first prove that there exist two normalized solutions if $q\in
(\frac{N+\alpha}{N},\frac{N+\alpha+2}{N})$ with $\mu >0$ small, that is, one is
at the negative energy level while the other one is at the positive energy
level.
  For $q=\frac{N+\alpha+2}{N}$, we show that there is a normalized ground state
for $0<\mu < \tilde{\mu} $ and there exist no ground states for $\mu
>\tilde{\mu}$, where $\tilde{\mu}$ is a sharp positive constant.
  If $q\in (\frac{N+\alpha+2}{N},\frac{N+\alpha}{N-2})$, we deduce that there
exists a normalized ground state for any $\mu>0$.
  We also obtain some existence and nonexistence results for the case $\mu<0$
and $q\in (\frac{N+\alpha}{N},\frac{N+\alpha+2}{N}]$.
  Besides, we analyze the asymptotic behavior of normalized ground states as
$\mu\rightarrow 0^{+}$.
