Let \(k\) be a real quadratic number field. If the \(2\)-Sylow subgroup of the class group of \(k\) has rank \(\geq 6\), then its \(2\)-class field tower is infinite by a famous result of E. S. Golod and I. R. Shafarevich [Transl., Ser. 2, Am. Math. Soc. 48, 91–102 (1965; Zbl 0148.28101)]. Various authors have studied (mostly imaginary) quadratic number fields whose class groups have \(2\)-rank just below the Golod-Shafarevich bound and have given sufficient criteria that guarantee that the \(2\)-class field tower is infinite (see e.g. E. Benjamin [Ann. Sci. Math. Qué. 26, No. 1, 1–13 (2002; Zbl 1033.11051], F. Hajir [Pac. J. Math. 176, No. 1, 15–18 (1996; Zbl 0879.11066)], C. Maire [Nagoya Math. J. 150, 1–11 (1998; Zbl 0923.11158)]). In this article, the author deals with the real quadratic case and presents results like the following: if the \(4\)-rank of the class group of \(k\) in the strict sense is \(\geq 3\), and if the \(2\)-rank of the class group in the usual sense equals \(5\), then the \(2\)-class field tower of \(k\) is infinite.