Study of the dephasing in electronic systems is not only important for probing the nature of their ground states, but also crucial to harnessing the quantum coherence for information processing. In contrast to well-studied conventional metals and semiconductors, it remains unclear which mechanism is mainly responsible for electron dephasing in three-dimensional topological insulators (TIs). Here, we report on using weak antilocalization effect to measure the dephasing rates in highly tunable (Bi,Sb)2Te3 thin films. As the transport is varied from a bulk-conducting regime to surface-dominant transport, the dephasing rate is observed to evolve from a linear temperature dependence to a sublinear power-law dependence. Although the former is consistent with the Nyquist electron-electron interactions commonly seen in ordinary 2D systems, the latter leads to enhanced electron dephasing at low temperatures and is attributed to the coupling between the surface states and the localized charge puddles in the bulk of 3D TIs.