In the quasi-linear approximation, we study electron acceleration process generated by whistler-mode and compressional ULF (fast mode waves) turbulences near the Earth's synchronous orbit. The results show that the whistler-mode turbulence (0.1fce ≤ f ≤ 0.75fce) can accelerate substorm injection electrons with several hundreds of keV through wave-particle gyroresonant interaction and hence may play an important role in the electron acceleration during substorms. The compressional ULF turbulence (2--15 mHz) can accelerate both lower-energy background electrons (<30 keV) and substorm injection electrons (~30--300 keV) through the transit-time damping mechanism. So the compressional ULF turbulence acceleration mechanism is important during both substorms and quiet times. The compressional ULF turbulence accelerates substorm injection electrons more effectively than whistler-mode turbulence. The combined electron acceleration by whistler-mode and ULF turbulences is most effective and can cause the number density of the relativistic electrons increase largely within about 8 hours. Substorms can offer both substorm injection electrons and strong turbulences, and therefore large flux enhancement events of relativistic electrons (≥1 MeV) always occur during substorm time. For magnetic storms that are composed of a series of substorms, extremely large flux enhancement events of the relativistic electrons can thus occur.