  Context. Weak lensing and clustering statistics beyond two-point functions
can capture non-Gaussian information about the matter density field, thereby
improving the constraints on cosmological parameters relative to the mainstream
methods based on correlation functions and power spectra. Aims. This paper
presents a cosmological analysis of the fourth data release of the Kilo Degree
Survey based on the density split statistics, which measures the mean shear
profiles around regions classified according to foreground densities. The
latter is constructed from a bright galaxy sample, which we further split into
red and blue samples, allowing us to probe their respective connection to the
underlying dark matter density. Methods. We use the state-of-the-art model of
the density splitting statistics and validate its robustness against mock data
infused with known systematic effects such as intrinsic galaxy alignment and
baryonic feedback. Results. After marginalising over the photometric redshift
uncertainty and the residual shear calibration bias, we measure for the full
KiDS-bright sample a structure growth parameter of $S_8 = \sigma_8
\sqrt{\Omega_\mathrm{m}/0.3} = 0.74^{+0.03}_{-0.02}$ that is competitive to and
consistent with two-point cosmic shear results, a matter density of
$\Omega_\mathrm{m} = 0.28 \pm 0.02$, and a constant galaxy bias of $b =
1.32^{+0.12}_{-0.10}$.
