The self-formation of incoherent laser pulses appears paradoxical, involving both a strong instability and a temporal localization process. Incoherent pulse regimes are nevertheless recurrent within ultrafast fiber laser dynamics. In this paper, we bring decisive experimental data, by recording in real time the buildup dynamics of incoherent pulses under different cavity dispersion regimes. Our measurements highlight different dominant mechanisms at play. Whereas soliton pulse shaping contributes to creating a chaotic bunch of pulses in the anomalous dispersion regime, incoherent pulses in the normal dispersion regime follow strongly turbulent dissipative dynamics. Numerical simulations reproduce qualitatively well the final buildup stage of observed dynamics. By displaying common dynamical features as well as disparities, these results support the development of a general concept of incoherent dissipative solitons. © 2020 authors. Published by the American Physical Society. Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.