In this paper we study strong approximation of the solution of a scalar stochastic differential equation (SDE) at the final time in the case when the drift coefficient may have discontinuities in space. Recently it has been shown in [Müller-Gronbach, T., and Yaroslavtseva, L., On the performance of the Euler-Maruyama scheme for SDEs with discontinuous drift coefficient, arXiv:1809.08423 (2018)] that for scalar SDEs with a piecewise Lipschitz drift coefficient and a Lipschitz diffusion coefficient that is non-zero at the discontinuity points of the drift coefficient the classical Euler-Maruyama scheme achieves an $L_p$-error rate of at least $1/2$ for all $p\in [1,\infty)$. Up to now this was the best $L_p$-error rate available in the literature for equations of that type. In the present paper we construct a method based on finitely many evaluations of the driving Brownian motion that even achieves an $L_p$-error rate of at least $3/4$ for all $p\in [1,\infty)$ under additional piecewise smoothness assumptions on the coefficients. To obtain this result we prove in particular that a quasi-Milstein scheme achieves an $L_p$-error rate of at least $3/4$ in the case of coefficients that are both Lipschitz continuous and piecewise differentiable with Lipschitz continuous derivatives, which is of interest in itself.