The diffusive motion of Brownian particles near irregular interfaces plays a crucial role in various transport phenomena in nature and industry. Most diffusion-reaction processes in confining inter- facial systems involve a sequence of Brownian flights in the bulk, connecting successive hits with the interface (Brownian bridges). The statistics of times and displacements separating two interface encounters are then determinant in the overall transport. We present a theoretical and numerical analysis of this complex first passage problem. We show that the bridge statistics is directly related to the Minkowski content of the surface within the usual diffusion length. In the case of self-similar or self-affine interfaces, we show and we check numerically that the bridge statistics follow power laws with exponents depending directly on the surface fractal dimension.