Aims. More than 30 years after its discovery, the thick disc of the Milky Way is not fully explored. We examine the shape of the thick disc in order to gain insight into the process of its formation. Methods. The shape of the thick disc is studied in detail using photometric data at high and intermediate latitudes from SDSS and 2MASS surveys. We adopted the population synthesis approach using an approximate Bayesian computation -Markov chain Monte Carlo (ABC-MCMC) method to determine the potential degeneracies in the parameters that can be caused by the mixing with the halo and the thin disc. We characterised the thick-disc shape, scale height, scale length, local density, and flare, and we investigated the extent of the thick-disc formation period by simulating several formation episodes. Results. We find that the vertical variation in density is not exponential, but much closer to a hyperbolic secant squared. Assuming a single formation epoch, the thick disc is better fitted with a sech 2 scale height of 470 pc and a scale length of 2.3 kpc. However, if one simulates two successive formation episodes, which mimicks an extended formation period, the older episode has a higher scale height and a longer scale length than the younger episode, which indicates a contraction during the collapse phase. The scale height decreases from 800 pc to 340 pc, the scale length from 3.2 kpc to 2 kpc. The likelihood is much higher when the thick disc formation extends over a longer period. We also show that star formation increases from the old episode to the young and that there is no flare in the outskirt of the thick disc during the main episode. We compare our results with formation scenarios of the thick disc. During the fitting process, the halo parameters are determined as well. If a power-law density is assumed, it has an exponent of 3.3 and an axis ratio of 0.7. Alternatively, a Hernquist shape would have an exponent of 2.76, an axis ratio of 0.77, and a core radius of 2.1 kpc.The constraint on the halo shows that a transition between an inner and outer halo, if it exists, cannot be at a distance shorter than about 30 kpc, which is the limit of our investigation using turnoff halo stars. Finally, we show that extrapolating the thick disc towards the bulge region explains well the stellar populations observed there that there is no longer need to invoke a classical bulge. Conclusions. The facts that the thick-disc episode lasted for several billion years, that a contraction is observed during the collapse phase, and that the main thick disc has a constant scale height with no flare argue against the formation of the thick disc through radial migration. The most probable scenario for the thick disc is that it formed while the Galaxy was gravitationally collapsing from well-mixed gas-rich giant clumps that were sustained by high turbulence, which prevented a thin disc from forming for a time, as proposed previously. This scenario explains well the observations in the thick-disc region and in the bulge region.
