Summary: A \(q\)-graph \(H\) on \(n\) vertices is a set of vectors of length \(n\) with all entries from \(\{0,1,\ldots,q\}\) and every vector (that we call a \(q\)-edge) having exactly two non-zero entries. The support of a \(q\)-edge \(\mathfrak{x}\) is the pair \(S_{\mathfrak{x}}\) of indices of non-zero entries. We say that \(H\) is an \(s\)-copy of an ordinary graph \(F\) if \(|H|=|E(F)|, F\) is isomorphic to the graph with edge set \(\{S_{\mathfrak{x}}:\mathfrak{x}\in H\}\), and whenever \(v\in e\), \(e^\prime \in E(F)\), the entries with index corresponding to \(v\) in the \(q\)-edges corresponding to \(e\) and \(e^\prime \) sum up to at least \(s\). E.g., the \(q\)-edges (1, 3, 0, 0, 0), (0, 1, 0, 0, 3), and (3, 0, 0, 0, 1) form a 4-triangle. The Turán number \(\mathrm{ex}(n,F,q,s)\) is the maximum number of \(q\)-edges that a \(q\)-graph \(H\) on \(n\) vertices can have if it does not contain any \(s\)-copies of \(F\). In the present paper, we determine the asymptotics of \(\mathrm{ex}(n,F,q,q+1)\) for many graphs \(F\).