The Galactic bulge is dominated by an old, metal-rich stellar population. The possible presence and the amount of a young (a few gigayears old) minor component is one of the major issues debated in the literature. Recently, the bulge stellar system Terzan 5 was found to harbor three sub-populations with iron content varying by more than one order of magnitude (from 0.2 up to two times the solar value), with chemical abundance patterns strikingly similar to those observed in bulge field stars. Here we report on the detection of two distinct main-sequence turnoff points in Terzan 5, providing the age of the two main stellar populations: 12 Gyr for the (dominant) sub-solar component and 4.5 Gyr for the component at super-solar metallicity. This discovery classifies Terzan 5 as a site in the Galactic bulge where multiple bursts of star formation occurred, thus suggesting a quite massive progenitor possibly resembling the giant clumps observed in star-forming galaxies at high redshifts. This connection opens a new route of investigation into the formation process and evolution of spheroids and their stellar content.