Ice bridges are static structures composed of tightly packed sea ice that can form during the course of its flow through a narrow strait. Despite their important role in local ecology and climate, the formation and breakup of ice bridges is not well understood and has proved difficult to predict. Using long-wave approximations and a continuum description of sea ice dynamics, we develop a one-dimensional theory for the wind-driven formation of ice bridges in narrow straits, which is verified against direct numerical simulations. We show that for a given wind stress and minimum and maximum channel widths, a steady-state ice bridge can only form beyond a critical value of the thickness and the compactness of the ice field. The theory also makes quantitative predictions for ice fluxes, which are particularly useful to estimate the ice export associated with the breakup of ice bridges. We note that similar ideas are applicable to dense granular flows in confined geometries.