  We establish new constraints on $f(T)$ gravity models by using cosmological
data. In particular, we investigate the restrictions given by the gas mass
fraction measurements of galaxy clusters and transversal BAO data. Both data
sets are regarded as weakly dependent on a fiducial cosmology. In addition, we
also include a CMB measurement of the temperature power spectrum first peak,
along with $H(z)$ values from cosmic chronometers and supernovae data from the
Pantheon data set. We also perform a forecast for future constraints on the
deviation of $f(T)$ models from the $\Lambda$CDM scenario by following the
specifications of the J-PAS and Euclid surveys and find significant
improvements on the constraints of the $b$-parameter, when compared to the
results of the statistical analysis.
