Abstract.In [J. Lie Theory 13 (2003), no. 2, 383–385], the authors introduced a framework to prove that a large class of HNN extensions have the Haagerup property, the main motivation being Baumslag–Solitar groups. Using this framework and new tools on locally compact groups developed in ["Amenable hyperbolic groups", preprint (2012)], we are able to obtain quantitative results on embeddings into Lebesgue spaces for a large class of HNN extensions.