  For the stationary nonlinear Schr\"odinger equation $-\Delta u+ V(x)u- f(u) =
\lambda u$ with periodic potential $V$ we study the existence and stability
properties of multibump solutions with prescribed $L^2$-norm. To this end we
introduce a new nondegeneracy condition and develop new superposition
techniques which allow to match the $L^2$-constraint. In this way we obtain the
existence of infinitely many geometrically distinct solutions to the stationary
problem. We then calculate the Morse index of these solutions with respect to
the restriction of the underlying energy functional to the associated
$L^2$-sphere, and we show their orbital instability with respect to the
Schr\"odinger flow. Our results apply in both, the mass-subcritical and the
mass-supercritical regime.
