Summary: Let \(\varepsilon \) be a real number such that \(0 < \varepsilon < \frac{1}{2}\) and \(t\) a positive integer. Let \(n\) be a sufficiently large positive integer as a function of \(t\) and \(\varepsilon \). We show that every \(n\)-vertex graph with at least \(n^{1+\varepsilon }\) edges contains a subdivision of \(K_{t}\) in which each edge of \(K_{t}\) is subdivided less than \(10/\epsilon \) times. This refines the main result in [\textit{A. Kostochka} and \textit{L. Pyber}, ``Small topological complete subgraphs of ``ense'' graphs,'' Combinatorica 8, No.\,1, 83--86 (1988; Zbl 0643.05039)] and resolves an open question raised there. We also pose some questions.