The temporal width of ultrashort electron pulses as a function of beam intensity was measured on the femtosecond time scale with a customized streak camera. The results show that the temporal profile of an electron pulse is Gaussian at low beam intensity and progressively evolves to a top-hat shape due to space charge broadening as the beam intensity increases. The strong correlation between the pulse width and beam intensity observed in our streaking measurements agrees very well with the mean-field calculation and supports the main conclusion of previous theoretical studies that the space charge broadening plays a determinant role.