Let \(G = (V, E)\) be a graph of order \(p\) and size \(q\). It is known that if \(G\) is a super edge-magic graph, then \(q \leq 2p – 3\). Furthermore, if \(G\) is super edge-magic and \(q = 2p – 3\), then the girth of \(G\) is \(3\). Additionally, if the girth of \(G\) is at least \(4\) and \(G\) is super edge-magic, then \(q \leq 2p – 5\). In this paper, we demonstrate that there are infinitely many graphs that are super edge-magic, have girth \(5\), and \(q = 2p – 5\). Hence, we conclude that for super edge-magic graphs of girths \(4\) and \(5\), the size is upper bounded by twice the order of the graph minus \(5\), and this bound is tight.