Phase-field fracture computations are based on constrained minimization of a non-convex energy functional depending on the displacement and damage fields. The alternate minimization procedure exploits the separate convexity of the energy with respect to the two fields by solving alternatively for each of them. However, due to strong non-linearities in both fields, convergence issues can affect the non-linear solvers used to determine the displacements and damage updates. Here, we address these issues adopting line-search-enhanced Newton solvers. Also, we exploit the separate convexity of the energy in the main fields to devise a simple yet robust line-search approach based on the energy derivative in direction of the Newton step. We compare its performance with other typical line-search approaches for two challenging benchmarks for the phase-field fracture model, namely the plate with a hole under compression and the Brazilian test. Results show that the proposed approach consistently performs among the best in comparison to other more classical approaches.