Prove convergence in $L^1$ if norms in $L^2$ are uniformly bounded
Notice that $\Vert f_n-f\Vert_2$ is bounded by $M'>0$. Then
$$ \int_E \vert f_n-f\vert d\mu\leq \Vert f_n-f\Vert_2\cdot \Big( \int_X \mathbf{1}_E^2d\mu \Big)^{\frac{1}{2}} =\Vert f_n-f\Vert_2\cdot \sqrt{\mu(E)}.$$