We report the measurement of the spin-orbit angle of the extra-solar planets HAT-P-8 b, HAT-P-9 b, HAT-P-16 b, and HAT-P-23 b, based on spectroscopic observations performed at the Observatoire de Haute-Provence with the SOPHIE spectrograph on the 1.93-m telescope. Radial velocity measurements of the Rossiter-McLaughlin effect show the detection of an apparent prograde, aligned orbit for all systems. The projected spin-orbit angles are found to be lambda = -17 degrees(+9.2)(-11.5), -16 degrees +/- 8 degrees, -10 degrees +/- 16 degrees, and +15 degrees +/- 22 degrees for HAT-P-8, HAT-P-9, HAT-P-16, and HAT-P-23, respectively, with corresponding projected rotational velocities of 14.5 +/- 0.8, 12.5 +/- 1.8, 3.9 +/- 0.8, and 7.8 +/- 1.6 km s(-1). These new results increase to 37 the number of accurately measured spin-orbit angles in transiting extrasolar systems. We conclude by drawing a tentative picture of the global behaviour of orbital alignement, involving the complexity and diversity of possible mechanisms.