This dataset includes experimental and numerical data from the manuscript 'Towards terawatt-scale spectrally-tunable terahertz pulses via relativistic laser-foil interactions'. The experimental data are obtained in the experiment titled “Intense terahertz radiation from picosecond laser-produced plasmas” (Application No.: 16110035, PI: David Neely), performed at Vulcan TAW. In this experiment, we demonstrate efficient production of terawatt (TW)-level THz pulses from high-intensity picosecond laser-foil interactions, and find the THz spectrum can be manipulated effectively by tuning the laser pulse duration or target size. A general analytical framework for THz generation is developed involving both the high-current electron emission and a time-varying electron sheath at the target rear, and the spectral tunability is found to stem from the change of the dominant THz generation mechanism. The THz radiation is applied to quantitatively diagnose the electron emission and the transient sheath, in good agreements with experimental measurements. Please see the file named “Data description” in the root directory of the attached zip file for a detailed description of the data contained therein. The data acquisition and analysis methods can also be found in the manuscript.