We develop a new interior point method for solving linear programs. Our algorithm is universal in the sense that it matches the number of iterations of any interior point method that uses a self-concordant barrier function up to a factor O(n1.5 log n) for an n-variable linear program in standard form. The running time bounds of interior point methods depend on bit-complexity or condition measures that can be unbounded in the problem dimension. This is in contrast with the simplex method that always admits an exponential bound. Our algorithm also admits a combinatorial upper bound, terminating with an exact solution in O(2n n1.5 log n) iterations. This complements previous work by Allamigeon, Benchimol, Gaubert, and Joswig (SIAGA 2018) that exhibited a family of instances where any path-following method must take exponentially many iterations.