  Current constraints on gravity are relatively weak on galactic and
intergalactic scales. Screened modified gravity models can exhibit complex
behaviour there without violating stringent tests of gravity within our Solar
System. They might hence provide viable extensions of the theory of gravity.
Here, we use galaxy kinematics to constrain screened modified gravity models.
We focus on chameleon $f(R)$ gravity and predict its impact on galaxy rotation
curves and radial acceleration relations. This is achieved by post-processing
state-of-the-art galaxy formation simulations from the \textsc{auriga project},
using the \textsc{mg-gadget} code. For a given galaxy, the surface dividing
screened and un-screened regions adopts an oblate shape, reflecting the disc
morphology of the galaxy's mass distribution. At the `screening radius'---the
radius at which screening is triggered in the disc plane---characteristic
`upturns' are present in both rotation curves and radial acceleration
relations. The locations of these features depend on various factors, such as
the galaxy mass, the concentration of the density profile and the value of the
background field amplitude $f_{R0}$. Self-screening of stars and environmental
screening also play a role. For Milky Way-size galaxies, we find that a model
with $|f_{R0}|=10^{-7}$ results in rotation curves that are indistinguishable
from $\Lambda$CDM, while for $|f_{R0}| \geq 2 \times 10^{-6}$ the simulated
galaxies are entirely unscreened, violating Solar System constraints. For
intermediate values, distinct upturns are present. With a careful statistical
analysis of existing samples of observed rotation curves, including lower mass
objects, constraints on $f(R)$ gravity with a sensitivity down to
$|f_{R0}|\sim10^{-7}$ should be possible.
