Context. Since their first detection in the interestellar medium, (sub-)millimeter line observations of different CO isotopic variants have routinely been employed to characterize the kinematic properties of the gas in molecular clouds. Many of these lines exhibit broad linewidths that greatly exceed the thermal broadening expected for the low temperatures found within these objects. These observed suprathermal CO linewidths are assumed to originate from unresolved supersonic motions inside clouds. Aims. The lowest rotational J transitions of some of the most abundant CO isotopologues, 12 CO and 13 CO, are found to present large optical depths. In addition to well-known line saturation effects, these large opacities present a non-negligible contribution to their observed linewidths. Typically overlooked in the literature, in this paper we aim to quantify the impact of these opacity broadening effects on the current interpretation of the CO suprathermal line profiles. Methods. Combining large-scale observations and LTE modeling of the ground J = 1−0 transitions of the main 12 CO, 13 CO, C 18 O isotopologues, we have investigated the correlation of the observed linewidths as a function of the line opacity in different regions of the Taurus molecular cloud. Results. Without any additional contributions to the gas velocity field, a large fraction of the apparently supersonic (M ∼ 2-3) linewidths measured in both 12 CO and 13 CO (J = 1−0) lines can be explained by the saturation of their corresponding sonic-like, optically thin C 18 O counterparts assuming standard isotopic fractionation. Combined with the presence of multiple components detected in some of our C 18 O spectra, these opacity effects also seem to be responsible for most of the highly supersonic linewidths (M > 8-10) detected in some of the broadest 12 CO and 13 CO spectra in Taurus.Conclusions. Our results demonstrate that most of the suprathermal 12 CO and 13 CO linewidths reported in nearby clouds like Taurus could be primarily created by a combination of opacity broadening effects and multiple gas velocity components blended in these saturated emission lines. Once corrected by their corresponding optical depth, each of these gas components present transonic intrinsic linewidths consistently traced by the three isotopologues, 12 CO, 13 CO, and C 18 O, with differences within a factor of 2. Highly correlated and velocity-coherent at large scales, the largest and highly supersonic velocity differences inside clouds are generated by the relative motions between individual gas components. In contrast to the classical interpretation within the framework of microscopic turbulence, this highly discretized structure of the molecular gas traced in CO suggest that the gas dynamics inside molecular clouds could be better described by the properties of a fully resolved macroscopic turbulence.