We study large random matrices with i.i.d. entries conditioned to have prescribed row and column sums (margins), a problem connected to relative entropy minimization, Schrödinger bridges, contingency tables, and random graphs with given degree sequences. Our central result is a `transference principle': the complex margin-conditioned matrix can be closely approximated by a simpler matrix whose entries are independent and drawn from an exponential tilting of the original model. The tilt parameters are determined by the sum of two potentials. We establish phase diagrams for `tame margins', where these potentials are uniformly bounded. This framework resolves a 2011 conjecture by Chatterjee, Diaconis, and Sly on $δ$-tame degree sequences and generalizes a sharp phase transition in contingency tables obtained by Dittmer, Lyu, and Pak in 2020. For tame margins, we show that a generalized Sinkhorn algorithm can compute the potentials at a dimension-free exponential rate. Our limit theory further establishes that for a convergent sequence of tame margins, the potentials converge as fast as the margins converge. We apply this framework and obtain several key results for the conditioned matrix: The marginal distribution of any single entry is asymptotically an exponential tilting of the base measure, resolving a 2010 conjecture by Barvinok on contingency tables. The conditioned matrix concentrates in cut norm around a `typical table' (the expectation of the tilted model), which acts as a static Schrödinger bridge between the margins. The empirical singular value distribution of the rescaled matrix converges to an explicit law determined by the variance profile of the tilted model. In particular, we confirm the universality of the Marchenko-Pastur law for constant linear margins.