We investigate the Tsallis holographic dark energy (THDE) models in the context of perturbations growth. We assume the description of dark energy by considering the holographic principle and the nonadditive entropy to carry out this. We implement the perturbed relativistic equations to achieve the growth of matter fluctuations, being the growth rate of the cosmic structures is non-negligible at low redshifts. To constrain and compare the models, we carry out the Bayesian analysis using the recent geometrical and growth rate observational data. The main results are: (i) the models are compatible with cosmological observations, (ii) the cosmological constant recovered with a $1sigma$ confidence level, furthermore (iii) they could cross the phantom barrier. Finally, the models can relieve $approx 1sigma$ the $sigma_8$ tension in the non-clustered case and can alleviate in $approx 2.8sigma$ the $H_0$ tension. From the model selection viewpoint, the data discarded the THDE models.
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