  We investigate the cosmological predictions of several $f(T)$ models, with up
to two parameters, at both the background and the perturbation levels. Using
current cosmological observations (geometric supernovae type Ia, cosmic
microwave background and baryonic acoustic oscillation and dynamical growth
data) we impose constraints on the distortion parameter, which quantifies the
deviation of these models from the concordance $\Lambda$ cosmology at the
background level. In addition we constrain the growth index $\gamma$ predicted
in the context of these models using the latest perturbation growth data in the
context of three parametrizations for $\gamma$. The evolution of the best fit
effective Newton constant, which incorporates the $f(T)$-gravity effects, is
also obtained along with the corresponding $1\sigma$ error regions. We show
that all the viable parameter sectors of the $f(T)$ gravity models considered
practically reduce these models to $\Lambda$CDM. Thus, the degrees of freedom
that open up to $\Lambda$CDM in the context of $f(T)$ gravity models are not
utilized by the cosmological data leading to an overall disfavor of these
models.
