A space-time approach is proposed to simulate the propagation of brittle cracks in an isotropic and elastic solid in dynamics. We adopt the so called phase-field description of crack that is based on a variational representation of fracture mechanics. Due to this variational structure, the crack initiation and propagation can be then described thanks to a well chosen potential. In this approach, we propose to consider a space-time potential to derive the appropriate Euler equations on the space-time domain. A time discontinuous Galerkin approach is used and adapted to damage and elastodynamics such as to be able to account of time singularities in the considered fields. This approach follows a previous work done on elastodynamics (see [47]) in which we have proposed a stabilized formulation with the help of least square terms. The proposed space-time potential is discretized with either standard finite-elements (ST-FE) or isogeometric analysis (ST-IGA). We apply this approach to different numerical examples including dynamic fragmentation.