Transverse beam coupling impedance is a source of beam instabilities that limits the machine performance in circular accelerators. Several beam based techniques have been used to measure the transverse impedance of an accelerator, usually based on the optics distortion produced by the impedance source itself. Beam position monitor turn-by-turn analysis for impedance characterization has been usually employed in large circumference machines, while synchrotron light sources have mainly used slow orbit based techniques. Instead, the work presented in this paper uses for the first time turn-by-turn data at ALBA to advance the measurement technique into the range of the typically small impedance values of modern light sources. We have measured local impedance contributions through the observation of phase advance versus bunch charge using the betatron oscillations excited with a fast dipole kicker. The ALBA beam position monitor system and the precision of the turn-by-turn analysis allowed to characterize the main sources of transverse impedance, in good agreement with the model values, including the impedance of an in-vacuum undulator.