Summary: Let \(\mathcal{F}\) be a family of graphs. The Turán number \(\operatorname{ex}( n ; \mathcal{F} )\) is defined to be the maximum number of edges in a graph of order \(n\) that is \(\mathcal{F} \)-free. P. Erdős and T. Gallai [Acta Math. Acad. Sci. Hung. 10, 337–356 (1959; Zbl 0090.39401)] determined the Turán number of \(M_{k + 1} \) (a matching of size \(k + 1)\) as follows: \[ \operatorname{ex}( n ; M_{k + 1} ) = \max \left\{ \binom{ 2 k + 1}{ 2}, \binom{ n}{ 2} - \binom{ n - k}{ 2}\right\}.\] Since then, there has been a lot of research on Turán number of linear forests. A linear forest is a graph whose connected components are all paths or isolated vertices. Let \(\mathcal{L}_{n , k}\) be the family of all linear forests of order \(n\) with \(k\) edges. In this paper, we prove that \[\operatorname{ex} ( n ; \mathcal{L}_{n , k} ) = \max \left\{ \binom{ k}{ 2}, \binom{ n }{2} - \binom{ n - \left\lfloor \frac{ k - 1}{ 2}\right\rfloor}{ 2} + c\right\},\] where \(c = 0\) if \(k\) is odd and \(c = 1\) otherwise. This determines the maximum number of edges in a non-Hamiltonian graph with given Hamiltonian completion number and also solves two open problems in [J. Wang and W. Yang, Discrete Appl. Math. 254, 291–294 (2019; Zbl 1404.05091)] as special cases. Moreover, we show that our main theorem implies Erdős-Gallai theorem and also gives a short new proof for it by the closure and counting techniques. Finally, we generalize our theorem to a conjecture which implies the famous Erdős matching conjecture.