Transport processes around the magnetic X-point of tokamaks, such as turbulence and mean-field drifts, are scarcely understood and difficult to investigate in experiments. In this paper, we explore the dynamics in a newly developed X-point scenario on the basic toroidal plasma device TORPEX and use it to validate plasma edge turbulence codes. In-situ measurements across the entire cross-section of TORPEX show turbulent structures forming in the vicinity of the X-point and being transported to the low-field side by small-scale fluctuations as well as background drifts. Full-scale simulations of this scenario are performed with the four state-of-the-art 3D fluid turbulence codes FELTOR, GBS, GRILLIX and STORM. The codes are validated through a rigorous procedure against time-averaged and fluctuation data obtained with high-resolution Langmuir probe arrays. We find that the codes are able to reproduce qualitatively, and in some cases quantitatively, some key features of the time-averaged fields, such as the radial fall-off length at mid-height and the background ExB flow pattern. The fluctuation levels, instead, are generally underestimated by typically factors of 2 or more. The sensitivity of the simulation results on the plasma collisionality and on the position of the sources, the input parameters with the largest uncertainty, is tested in GBS, showing a mild effect on the overall quantitative agreement with the experiment. Overall, this validation reveals the challenges to reproduce the plasma dynamics near an X-point. The application of this systematic validation procedure will allow the impact of future developments in the codes to be assessed objectively.