The James Webb Space Telescope provides an exciting opportunity to investigate stellar systems located in heavily obscured regions like the Galactic bulge. Possibly the most enigmatic among them is Terzan 5: long classified as a globular cluster, it is now known to host distinct stellar populations with different iron abundances, ranging approximately from [Fe/H]=-0.8 to [Fe/H]=+0.3dex. Indeed, the chemical and structural properties collected so far suggest that it is the remnant of one of the primordial clumps that contributed to the early assembly of the bulge, a so-called "Bulge Fossil Fragment". Here we present a new photometric analysis of Terzan 5 based on JWST/NIRCam observations in the F115W and F200W filters, as well as archival HST/ACS optical (F606W and F814W) data. The dataset overcomes the severe and spatially variable extinction along the line of sight and yields the deepest color-magnitude diagram ever obtained for Terzan 5. Proper motion selections and high-resolution differential reddening corrections allow us to isolate bona fide cluster members and to provide an unprecedented view of the main-sequence turn-off region. We clearly identify two main components and determine their respective ages: the old, sub-solar component has an age of 12.5+/-0.5Gyr, while the super-solar component is significantly younger, with an age of 4.7+/-0.5Gyr. Interestingly, we also find hints of an even younger main-sequence turn-off and sub-giant branch, consistent with the presence of a further stellar component with an age of 3.8+/-0.5Gyr. There is also evidence of a blue plume populated by stars as bright as m(F115W)~17.4, suggesting a prolonged period of star formation extending up to 2.5Gyr ago.