Topology optimization, i.e. the optimal distribution of material in order to fulfil a given goal, can reduce the needed material for a structure, thereby saving energy consumption and natural resources. However, usually the nonlinear material behaviour, such as crack initiation and propagation, is not considered during the optimization process. This may lead to optimized designs that are susceptible to fracture. In recent years, the application of topology optimization for enhancing the resistance of a structure to fracture has gained increasing interest. In this contribution, topology optimization is applied for shells in order to enhance their resistance to brittle fracture. In order to reduce the computational cost, an isogeometric Reissner-Mindlin formulation is applied that considers the shell kinematics only on the mid-surface of the structure. This is important for the combination with topology optimization, since the nonlinear brittle fracture problem must be solved in each optimization step. A phase-field model for brittle fracture is used to describe crack initiation and propagation in the shell structure. The brittle fracture resistance of the shell is increased by applying a topology optimization process that maximizes the external work while considering the volume fraction constraint. Furthermore, the shell formulation is combined with isogeometric analysis that uses higher-order Non-Uniform Rational B-Spline (NURBS) basis functions. These are characterized by high continuity and the ability to naturally eliminate numerical instabilities in the topology optimization process. In this case, both the displacement/rotations and the phase-field are discretized using NURBS basis functions. This additionally offers the advantage that the design, topology optimization, structural analysis process, and post-processing for additive manufacturing are unified since NURBS basis functions are commonly included in CAD tools as well. Various numerical examples are presented in order to verify the accuracy and effectiveness of the method.