  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
$1\sigma$ confidence level, furthermore (iii) they could cross the phantom
barrier. Finally, the models can relieve $\approx 1\sigma$ the $\sigma_8$
tension in the non-clustered case and can alleviate in $\approx 2.8\sigma$ the
$H_0$ tension. From the model selection viewpoint, the data discarded the THDE
models.
