  We investigate galactic rotation curves in $f(T)$ gravity, where $T$
represents a torsional quantity. Our study centers on the particular Lagrangian
$f(T)=T+\alpha{T^n}$, where $|n|\neq 1$ and $\alpha$ is a small unknown
constant. To do this we treat galactic rotation curves as being composed from
two distinct features of galaxies, namely the disk and the bulge. This process
is carried out for several values of the index $n$. The resulting curve is then
compared with Milky Way profile data to constrain the value of the index $n$
while fitting for the parameter $\alpha$. These values are then further tested
on three other galaxies with different morphologies. On the galactic scale we
find that $f(T)$ gravity departs from standard Newtonian theory in an important
way. For a small range of values of $n$ we find good agreement with data
without the need for exotic matter components to be introduced.
