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An ever-increasing number of strong-field applications, such as ultrafast coherent control over matter and light, require driver light pulses that are both high power and spectrally tunable. The realization of such a source in the terahertz (THz) band has long been a formidable challenge. Here, we demonstrate, via experiment and theory, efficient production of terawatt (TW)-level THz pulses from high-intensity picosecond laser irradiation on a metal foil. It is shown that 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. In addition to being an ultrabright source (brightness temperature of about 10^{21} K) for extreme THz science, the THz radiation presented here also enables a unique in situ laser-plasma diagnostic. Employing the THz radiation to quantify the escaping electrons and the transient sheath shows good agreement with experimental measurements.