  The strong rate of convergence of the Euler-Maruyama scheme for nondegenerate
SDEs with irregular drift coefficients is considered. In the case of
$\alpha$-H\"older drift in the recent literature the rate $\alpha/2$ was proved
in many related situations. By exploiting the regularising effect of the noise
more efficiently, we show that the rate is in fact arbitrarily close to $1/2$
for all $\alpha>0$. The result extends to Dini continuous coefficients, while
in $d=1$ also to all bounded measurable coefficients.
