The Schrödinger Bridge (SB) has recently emerged as an important optimization problem on the space of probability measures that underlies the development of potent AI frameworks like diffusion models. Despite its significance, devising an efficient algorithmic solution remains a challenging task. To this end, we introduce a novel geometric framework for optimizing SB: a continuous-time adaptation of the Sinkhorn algorithm. This innovation yields novel Sinkhorn variants equipped with variable step sizes, offering a crucial advantage over existing methods by guaranteeing convergence even in the presence of noise and bias. Central to our approach is a novel Riemannian geometric structure tailored for probability measures, extending the classical Fisher-Rao metric to encompass conditional variants. Furthermore, via exploiting its intimate connection with the theory of mirror descent, our methodology leads to an accelerated variant of Sinkhorn dynamics.