Context. Investigation of the formation, origin, and evolution of the dichotomy of the Milky Way's thin and thick disc components has been a focal point of research since it is key to understanding the formation of our Galaxy. One difficulty in this pursuit is that the populations defined based on their morphology or kinematics show a mix of chemically distinct populations. Age is then a key parameter to understand the disc evolution. Aims. We aim to derive age and metallicity distributions of the kinematic thick and thin discs in order to reveal details of the duration, intensity, and relation between the star formation episodes that led to the current kinematic thick-thin disc configuration. Methods. We applied the CMDft.Gaia pipeline based on a colour-magnitude diagram fitting technique to derive the dynamically evolved star formation history (deSFH) of the kinematically selected thin and thick discs. The analysis is based on Gaia DR3 data within a cylindrical volume centred on the Sun with a radius of 250 pc and a height of 1 kpc. Results. Our analysis shows that the kinematically selected thick disc is predominantly older than 10 Gyr and underwent a rapid metallicity enrichment through three main episodes. The first occurred over 12 Gyr ago, peaking at [Z/H] ∼−0.5 dex; the second was around 11 Gyr ago and caused a rapid increase in metallicity up to [Z/H]=0.0 and a broad spread in [α/Fe] from ∼ 0.3 to solar values; and the third, just over 10 Gyr ago, reached super-solar metallicities. In contrast, the kinematic thin disc stars began forming about 10 Gyr ago, coinciding with the thick disc's star formation end, and is characterised by a super-solar metallicity and low [α/Fe]. The transition between the kinematic thick and thin discs aligns with the Milky Way's last major merger: the accretion of Gaia-Sausage-Enceladus (GSE). We also identify a small population of kinematically selected thin disc stars with high and intermediate-[α/Fe] abundances, slightly older than 10 billion years, indicating a kinematic transition from thick to thin disc during the Milky Way's high and intermediate- [α/Fe] phase. The kinematic thin disc's age-metallicity relation reveals overlapping star formation episodes with distinct metallicities, suggesting radial mixing in the solar neighbourhood, with the greatest spread around 6 Gyr ago. Additionally, we detect an isolated thick disc star formation event 6 Gyr ago at solar metallicity, and it coincides with the estimated first pericentre of the Sagittarius satellite galaxy.