The non-trivial topology of three-dimensional topological insulators dictates the appearance of gapless Dirac surface states. Intriguingly, when made into a nanowire, quantum confinement leads to a peculiar gapped Dirac sub-band structure. This gap is useful for, e.g., future Majorana qubits based on TIs. Furthermore, these sub-bands can be manipulated by a magnetic flux and are an ideal platform for generating stable Majorana zero modes, playing a key role in topological quantum computing. However, direct evidence for the Dirac sub-bands in TI nanowires has not been reported so far. Here, using devices fabricated from thin bulk-insulating (Bi1-xSbx)2Te3 nanowires we show that non-equidistant resistance peaks, observed upon gate-tuning the chemical potential across the Dirac point, are the unique signatures of the quantized sub-bands. These TI nanowires open the way to address the topological mesoscopic physics, and eventually the Majorana physics when proximitized by an s-wave superconductor.