To investigate the origin of the features discovered in the exoplanet population, the knowledge of exoplanets' mass and radius with a good precision (≲10%) is essential. To achieve this purpose the discovery of transiting exoplanets around bright stars is of prime interest. In this paper, we report the discovery of three transiting exoplanets by the SuperWASP survey and the SOPHIE spectrograph with mass and radius determined with a precision better than 15%. WASP-151b and WASP-153b are two hot Saturns with masses, radii, densities and equilibrium temperatures of 0.31_−0.03^+0.04 M_J, 1.13_−0.03^+0.03 R_J, 0.22_−0.02^+0.03 ρ_J and 1290₋₁₀⁺²⁰ K, and 0.39_−0.02^+0.02 M_J, 1.55_−0.08^+0.10 R_J, 0.11_−0.02^+0.02 ρ_J and 1700_−0.40^+0.40 K, respectively. Their host stars are early G type stars (with mag V ~ 13) and their orbital periods are 4.53 and 3.33 days, respectively. WASP-156b is a super-Neptune orbiting a K type star (mag V = 11.6). It has a mass of 0.128_−0.009^+0.010 M_J, a radius of 0.51_−0.02^+0.02 R_J, a density of 1.0_−0.1^+0.1 ρ_J, an equilibrium temperature of 970₋₂₀⁺³⁰ K and an orbital period of 3.83 days. The radius of WASP-151b appears to be only slightly inflated, while WASP-153b presents a significant radius anomaly compared to a recently published model. WASP-156b, being one of the few well characterized super-Neptunes, will help to constrain the still debated formation of Neptune size planets and the transition between gas and ice giants. The estimates of the age of these three stars confirms an already observed tendency for some stars to have gyrochronological ages significantly lower than their isochronal ages. We propose that high eccentricity migration could partially explain this behavior for stars hosting a short period planet. Finally, these three planets also lie close to (WASP-151b and WASP-153b) or below (WASP-156b) the upper boundary of the Neptunian desert. Their characteristics support that the ultra-violet irradiation plays an important role in this depletion of planets observed in the exoplanet population.