Recent studies of gas kinematics at high redshift have reported disky systemswhich appear to challenge models of galaxy formation, but it is unclear whetherthey are representative of the underlying galaxy population. We present thefirst statistical sample of spatially resolved ionised gas kinematics at highredshift, comprised of $272$ H$\alpha$ emitters in GOODS-S and GOODS-N atredshifts $z\approx3.9-6.5$, observed with JWST/NIRCam slitless spectroscopyand imaging from JADES, FRESCO and CONGRESS. The sample probes two orders ofmagnitude in stellar mass ($\log (M_{\star}[\mathrm{M}_{\odot}])\approx8-10$)and star formation rate ($\text{SFR}\approx0.3-100\thinspace M_{\odot}/$yr),and is representative down to $\log(M_{\star}[\mathrm{M}_{\odot}])\approx 9$.Using a novel inference tool, $\texttt{geko}$, we model the grism data tomeasure morphological and kinematic properties of the ionised gas, as probed byH$\alpha$. Our results are consistent with a decrease of the rotational support$v/\sigma_0$\ and increase of the velocity dispersion $\sigma_0$ with redshift,with $\sigma_0\approx100$ km/s and $v/\sigma_0\approx1-2$ at $z\approx3.9-6.5$.We study the relations between $\sigma_0$, and $v/\sigma_0$, and different starformation tracers and find a large scatter and diversity, with the strongestcorrelations between $\sigma_0$ and SFR and SFR surface density. The fractionof rotationally supported systems ($v/\sigma_0>1$) slightly increases withcosmic time, from $(36\pm6)\%$ to $(41\pm6)\%$ from $z\sim 5.5$ to $z\sim 4.5$,for galaxies with masses $9<\log(M_{\star}[\mathrm{M}_{\odot}])<10$. Overall,disks do not dominate the turbulent high-redshift galaxy population in the massrange probed by this work. When placed in the context of studies up to cosmicnoon, our results are consistent with a significant increase of disk-likesystems with cosmic time.