We relax the marginal constraints of the classical Schrödinger bridge by penalizing the transport cost between the marginals of the bridge and the prescribed marginals. We derive a duality formula for such transport-relaxed Schrödinger bridge and show that it reduces to a finite-dimensional concave optimization problem when the prescribed marginals are discrete and the reference distribution is absolutely continuous. Existence and uniqueness are established for both the primal and dual problems. Moreover, as the penalty goes to infinite, we characterize the limiting bridge as the solution to a discrete Schrödinger bridge and identify a logarithmic leading-order blow-up. Finally, we propose a gradient ascent algorithm and a Sinkhorn-type algorithm to numerically solve the transport-relaxed Schrödinger bridge. We obtain a linear convergence rate for both algorithms.