We address the problem of visually guided rearrangement planning with many movable objects, i.e., finding a sequence of actions to move a set of objects from an initial arrangement to a desired one, while relying on visual inputs coming from an RGB camera. To do so, we introduce a complete pipeline relying on two key contributions. First, we introduce an efficient and scalable rearrangement planning method, based on a Monte-Carlo Tree Search exploration strategy. We demonstrate that because of its good trade-off between exploration and exploitation our method (i) scales well with the number of objects while (ii) finding solutions which require a smaller number of moves compared to the other state-of-the-art approaches. Note that on the contrary to many approaches, we do not require any buffer space to be available. Second, to precisely localize movable objects in the scene, we develop an integrated approach for robust multi-object workspace state estimation from a single uncalibrated RGB camera using a deep neural network trained only with synthetic data. We validate our multi-object visually guided manipulation pipeline with several experiments on a real UR-5 robotic arm by solving various rearrangement planning instances, requiring only 60 ms to compute the plan to rearrange 25 objects. In addition, we show that our system is insensitive to camera movements and can successfully recover from external perturbations. Supplementary video, source code and pre-trained models are available at https://ylabbe.github.io/rearrangement-planning/.