This paper addresses the problem of minimizing multilabel energies with truncated convex priors. Such priors are known to be useful but difficult and slow to optimize because they are not convex. We propose two novel classes of binary Graph-Cuts (GC) moves, namely the convex move and the quantized move. The moves are complementary. To significantly improve efficiency, the label range is divided into even intervals. The quantized move tends to efficiently put pixel labels into the correct intervals for the energy with truncated convex prior. Then the convex move assigns the labels more precisely within these intervals for the same energy. The quantized move is a modified α-expansion move, adapted to handle a generalized Potts prior, which assigns a constant penalty to arguments above some threshold. Our convex move is a GC representation of the efficient Murota's algorithm. We assume that the data terms are convex, since this is a requirement for Murota's algorithm. We introduce Quantized-Convex Split Moves algorithm which minimizes energies with truncated priors by alternating both moves. This algorithm is a fast solver for labeling problems with a high number of labels and convex data terms. We illustrate its performance on image restoration.