AbstractWe show that$$\mathcal {M}_{g,n}$$Mg,n, the moduli space of smooth curves of genusgtogether withnmarked points, is unirational for$$g=12$$g=12and$$2 \le n\le 4$$2≤n≤4and for$$g=13$$g=13and$$1 \le n \le 3$$1≤n≤3, by constructing suitable dominant families of projective curves in$$\mathbb {P}^1 \times \mathbb {P}^2$$P1×P2and$$\mathbb {P}^3$$P3respectively. We also exhibit several new unirationality results for moduli spaces of smooth curves of genusgtogether withnunordered points, establishing their unirationality for$$g=11, n=7$$g=11,n=7and$$g=12, n =5,6$$g=12,n=5,6.