Context. About 35% of the nearby disc galaxies host a weak bar for which different formation scenarios, including the weakening of a strong bar and tidal interaction with a companion, have been suggested. Measuring the bar pattern speeds of a sample of weakly barred galaxies is a key step in constraining their formation process, but such a systematic investigation is still missing. Aims. We investigated the formation process of weak bars by measuring their properties in a sample of 29 nearby weakly barred galaxies, spanning a wide range of morphological types and luminosities. The sample galaxies were selected to have an intermediate inclination, a bar at an intermediate angle between the disc minor and major axes, and an undisturbed morphology and kinematics to allow the direct measurement of the bar pattern speed. Combining our analysis with previous studies, we compared the properties of weak and strong bars. Methods. We measured the bar radius and strength from the r band images available in the Sloan Digital Sky Survey and bar pattern speed and corotation radius from the stellar kinematics obtained by the Calar Alto Legacy Integral Field Area Survey. We derived the bar rotation rate as the ratio between the corotation and bar radii. Results. Thirteen out of 29 galaxies (45%), which were morphologically classified as weakly barred from a visual inspection, do not actually host a bar component or their central elongated component is not in rigid rotation. We successfully derived the bar pattern speed in 16 objects. Two of them host an ultrafast bar. Using the bar strength to differentiate between weak and strong bars, we found that the weakly barred galaxies host shorter bars with smaller corotation radii than their strongly barred counterparts. Weak and strong bars have similar bar pattern speeds and rotation rates, which are all consistent with being fast. We did not observe any difference between the bulge prominence in weakly and strongly barred galaxies, whereas nearly all the weak bars reside in the disc inner parts, contrary to strong bars. Conclusions. We ruled out that the bar weakening is only related to the bulge prominence and that the formation of weak bars is triggered by the tidal interaction with a companion. Our observational results suggest that weak bars may be evolved systems exchanging less angular momentum with other galactic components than strong bars.