We have experimentally investigated long-time ([gt]10 h) magnetic relaxation in single crystals of Bi[sub 2]Sr[sub 2]CaCu[sub 2]O[sub [ital x]] and YBa[sub 2]Cu[sub 8]O[sub [ital x]] at wide ranges of temperature and field. We have found that the flux motion in single crystals of high-[ital T][sub [ital c]] superconductors exhibits two distinctly different behaviors at a wide regime of driving force. At large driving force, the flux motion is dominated by thermally activated avalanches, which can be described by a so-called self-organized criticality theory. At small driving force, however, the avalanche effect is significantly reduced, and the flux motion is characterized by slow creep which can be described by the Anderson-Kim model. Therefore, there is a dynamic crossover between the flux avalanches and a pure thermally activated regime in high-[ital T][sub [ital c]] superconductors.