Using multigraded Castelnuovo–Mumford regularity, we study the equations defining a projective embedding of a variety $X$. Given globally generated line bundles $B_{1}, \dotsc, B_{\ell}$ on $X$ and $m_{1}, \dotsc, m_{\ell} \in \mathbb{N}$, consider the line bundle $L := B_{1}^{m_{1}} \otimes \dotsb \otimes B_{\ell}^{m_{\ell}}$. We give conditions on the $m_{i}$ which guarantee that the ideal of $X$ in $\mathbb{P}(H^{0}(X,L)^{*})$ is generated by quadrics and that the first $p$ syzygies are linear. This yields new results on the syzygies of toric varieties and the normality of polytopes.