We present a new derivation of the star formation history (SFH) of the dSph galaxy Fornax in two central regions, characterized by unprecedented precision and age resolution. It reveals that star formation has proceeded in sharp bursts separated by periods of low level or quiescent activity. The SFH was derived through colour-magnitude diagram (CMD) fitting of two extremely deep Hubble Space Telescope CMDs, sampling the centre and one core radius. The attained age resolution allowed us to single out a major star formation episode at early times, a second strong burst 4.6 +/- 0.4 Gyr ago and recent intermittent episodes similar to 2-0.2 Gyr ago. Detailed testing with mock stellar populations was used to estimate the duration of the main bursts and study the occurrence of low-level star formation between them. The SFHs in both regions show common features, with activity at the same epochs and similar age-metallicity relationship. However, clear indications of a spatial gradient were also found, with mean age increasing with radius and star formation episodes being more prolonged in the centre. While some galaxy evolution models predict bursty SFHs in dwarf galaxies and thus a secular origin of the observed SFH cannot be excluded in Fornax, other evidence points to possible mergers or interactions as the cause of its bursty SFH. In particular, we calculated the Fornax orbit relative to the closest dwarfs and the Milky Way and observed a correspondence between the main intermediate-age and young events and peri-passages of Fornax around the Milky Way, possibly indicating tidally induced star formation.