  Peculiar velocities are an important probe of the mass distribution in the
Universe and the growth rate of structure, directly measuring the effects of
gravity on the largest scales and providing a test for theories of gravity.
Comparing peculiar velocities predicted from the density field mapped by a
galaxy redshift survey with peculiar velocities measured using a distance
estimator such as the Tully-Fisher relation yields the growth factor for
large-scale structure. We present a method for forward-modelling a sample of
galaxy magnitudes and velocity widths that simultaneously determines the
parameters of the Tully-Fisher relation and the peculiar velocity field. We
apply this to the Cosmicflows-4 (CF4) Tully-Fisher dataset, using the peculiar
velocities predicted from the 2M++ redshift survey. After validating the method
on mock surveys, we measure the product of the growth rate and mass fluctuation
amplitude to be $f\!\sigma_8$ = 0.35$\pm$0.03 at an effective redshift of $z$ =
0.017. This is consistent at 3$\sigma$ with the Planck CMB prediction, even
though the uncertainty does not fully account for all sources of sample
variance. We find the residual bulk flow from gravitational influences outside
the 2M++ survey volume to be $|V|$ = 227$\pm$11 km/s, $(l,b)$ =
(303$^\circ$,$-$1$^\circ$) in Galactic polar coordinates and the CMB frame.
Using simulations, we show that applying our methodology to the large new
sample of Tully-Fisher peculiar velocities expected from the WALLABY HI survey
of the southern sky can improve the constraints on the growth rate by a factor
of 2-3.
