We report anomalous enhancement of the critical current at low temperatures in gate-tunable Josephson junctions made from topological insulator BiSbTeSe_{2} 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}∝exp(-k_{B}T/δ), where k_{B} is the Boltzmann constant. The extracted characteristic energy scale δ 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 δ to low-energy Andreev bound states arising from winding of the electronic wave function around the circumference of the topological insulator nanoribbon.