In the present work we aim to simulate shallow water flows over movable bottom with suspended and bedload transport. In order to numerically approximate such a system, we proceed step by step. We start by considering shallow water equations with non-constant density of the mixture water-sediment. Then, the Exner equation is included to take into account bedload sediment transport. Finally, source terms for friction, erosion and deposition processes are considered. Indeed, observe that the sediment particle could go in suspension into the water or being deposited on the bottom. For the numerical scheme, we rely on well-balanced Lagrange-projection methods. In particular, since sediment transport is generally a slow process, we aim to develop semi-implicit schemes in order to obtain fast simulations. The Lagrange-projection splitting is well-suited for such a purpose as it entails a decomposition of the (fast) acoustic waves and the (slow) material waves of the model. Hence, in subsonic regimes, an implicit approximation of the acoustic equations allows us to neglect the corresponding CFL condition and to obtain fast numerical schemes with large time step.