The anomalous Hall (AH) and spin Hall effects are important tools for the generation, control, and detection of spin and spin-polarized currents in solids and, thus, hold promises for future spintronic applications. Despite tremendous work on these effects, their ultrafast dynamic response is still not well explored. Here, we induce ultrafast AH currents in a magnetically-biased semiconductor by optical femtosecond excitation at room temperature. The currents' dynamics are studied by detecting the simultaneously emitted THz radiation. We show that the temporal shape of the AH currents can be extracted by comparing its THz radiation to the THz radiation emitted from optically induced currents whose temporal shape is well known. We observe a complex temporal shape of the AH currents suggesting that different microscopic origins contribute to the current dynamics. This is further confirmed by photon energy dependent measurements revealing a current inversion at low optical excitation intensities. Our work is a first step towards full time resolution of AH and spin Hall currents and helps to better understand the underlying microscopic origins, being a prerequisite for ultrafast spintronic applications using such currents.