We design moving horizon state estimators for a general model of bioprocesses. The underlying optimization is nonconvex due to the microbial growth kinetics, which are modeled as nonlinear functions. We relax the nonconvex growth constraints so that the optimization becomes a second-order cone program, which can be solved efficiently at large scales. Unfortunately, solutions to the relaxation can be inexact and thus lead to inaccurate state estimates. To recover feasible, albeit potentially locally optimal solutions, we use the concave-convex procedure, which here takes the form of a sequence of second-order cone programs. We find that the moving horizon state estimators outperform the unscented Kalman filter on numerical examples based on the gradostat and anaerobic digestion when there is high process noise or parameter error.