The mixed-norm cost functions arise in many applied optimization problems. As an important example, we consider the state estimation problem for a linear dynamic system under a nonclassical assumption that some entries of state vector admit jumps in their trajectories. The estimation problem is solved by means of mixed l(1)/l(2)-norm approximation. This approach combines the advantages of the well-known quadratic smoothing and the robustness of the least absolute deviations method. For the implementation of the mixed-norm approximation, a dynamic iterative estimation algorithm is proposed. This algorithm is based on weight and time recursions and demonstrates the high efficiency. It well identifies the rare jumps in the state vector and has some advantages over more customary methods in the typical case of a large amount of measurements. Nonoptimality levels for current iterations of the algorithm are constructed. Computation of these levels allows to check the accuracy of iterations.