The behaviour of incompressible side-branching flows is examined theoretically at high Reynolds numbers and compared with direct numerical simulation at moderate Reynolds numbers. The theoretical model assumes the branching (daughter) tube is small compared to the main (mother) tube and that the branching angle is small. The theory is applicable to steady and unsteady flows in two or three dimensions, and to a broad range of flow splits between mother and daughter vessels. The first main result of the work is that, in the vicinity of the branch, the flow adjusts to the imposed downstream pressure in the daughter tube through a jump (a rapid change over a short length scale) in flow properties across the daughter entrance. It is shown that, for large pressure drops in the daughter tube, fluid is sucked in at high velocities from the mother and thereby provides a favourable upstream feedback. This counteracts the tendency of the flow to separate from what would otherwise be an adversely shaped upstream wall. Increased divergence of mother and daughter tubes can thus be achieved at high daughter flow rates without separation. The second main result of the work is that the direct numerical simulations confirm the very rapid variation in flow properties and show reasonable agreement with the theory at moderate Reynolds numbers.