We report on an instrument for time-resolved spectroscopy (TRS) based on white-light generation in a highly non-linear crystal fiber. TRS in the visible and near-infrared region at picosecond-to-nanosecond time scales has attracted increased interest in recent years owing to the possibility of spectroscopic analysis of turbid media, such as biological tissues. A self-mode-locked Ti:Sapphire oscillator pumped by an Ar:ion laser provides pulses 50 - 100 fs long, at 85 MHz repetition rate. The light is focused into a crystal fiber, which consists of a core surrounded by a mesh of air-filled holes. White light is generated by a combination of several non-linear effects in the fiber. We optimize the spectrum for measurements in the region 600 - 1000 nm. For detection, we use an imaging spectrometer coupled to a 16-channel photomultiplier tube, enabling simultaneous detection in 16 wavelength bands. We use time-correlated single-photon counting to record the signal, with a temporal resolution of similar to 160 ps. To demonstrate the system, we have performed measurements of the diffuse time-resolved reflectance of tissue phantoms made of epoxy resin with added scattering and absorbing materials. The data was evaluated using a light propagation model based on diffusion theory, to extract the scattering and absorption coefficients of the medium. The results corresponded very well with previous measurements on the phantoms performed using other TRS instruments.