  Three-dimensional conformal field theories (CFTs) with slightly broken higher
spin symmetry provide an interesting laboratory to study general properties of
CFTs and their roles in the AdS/CFT correspondence. In this work we compute the
four-point functions at arbitrary 't Hooft coupling $\lambda$ in the CFTs with
slightly broken higher spin symmetry. We use a bootstrap approach based on the
approximate higher spin Ward identity. We show that the bootstrap equation is
separated into two parts with opposite parity charges, and it leads to a
recursion relation for the $\lambda$ expansions of the correlation functions.
The $\lambda$ expansions terminate at order $\lambda^2$ and the solutions are
exact in $\lambda$. Our work generalizes the approach proposed by Maldacena and
Zhiboedov to four-point correlators, and it amounts to an on-shell study for
the $3D$ Chern-Simons vector models and the Vasiliev theory in $AdS_4$.
Besides, we show that the same results can also be obtained rather simply from
bosonization duality of $3D$ Chern-Simons vector models. The odd term at order
$O(\lambda)$ in the spinning four-point function relates to the free boson
correlator through a Legendre transformation. This provides new evidence on the
$3D$ bosonization duality at the spinning four-point function level. We expect
this work can be generalized to a complete classification of general four-point
functions of single trace currents.
