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Variations in scaling behaviour in the flux and emissions of gravitational lensed quasars can provide valuable information about the dynamics within the sources and their cosmological evolution with time. Here, we study the multifractal behaviour of the light curves (LCs) of 14 lensed quasars with multiple images in the r band, with redshift ranging from 0.657 to 2.730, in the search for potential differences in non-linearity between the signals of the quasar multiple images. Among these lensed systems, nine present two images, two present three images, and three present four images. To this end, we apply the wavelet transform-based multifractal analysis formalism called wavelet transform modulus maxima. We identify strong multifractal signatures in the LCs of the images of all analysed lensed quasar systems, independently of the number of images, with a significant difference between the degree of multifractality of all the images and combinations. We have also searched for a possible connection between the degree of multifractality and the characteristic parameters related to the quasar source and the lensing galaxy. These parameters include the Einstein ring radius and the accretion disc size and the characteristic time-scales related to microlensing variability. The analysis reveals some apparent trends, pointing to a decrease in the degree of multifractality with the increase of the quasar's source size and time-scale. Using a larger sample and following a similar approach, this study confirms a previous finding for the quasar Q0957 + 561. [ABSTRACT FROM AUTHOR]