We report anomalous enhancement of the critical current at low temperatures in gate-tunable Josephson junctions made from topological insulator BiSbTeSe2 nanoribbons with superconducting Nb electrodes. In contrast to conventional junctions, as a function of the decreasing temperature T, the increasing critical current I-c exhibits a sharp upturn at a temperature T-* around 20% of the junction critical temperature for several different samples and various gate voltages. The I-c vs T demonstrates a short junction behavior for T > T-*, but crosses over to a long junction behavior for T < T-* with an exponential T dependence I-c proportional to exp(-k(B)T/delta), where k(B) is the Boltzmann constant. The extracted characteristic energy scale delta is found to be an order of magnitude smaller than the induced superconducting gap of the junction. We attribute the long-junction behavior with such a small delta to low-energy Andreev bound states arising from winding of the electronic wave function around the circumference of the topological insulator nanoribbon.