  The Tur\'{a}n number of a graph $H$, $ex(n,H)$, is the maximum number of
edges in any graph of order $n$ which does not contain $H$ as a subgraph.
Lidick\'{y}, Liu and Palmer determined $ex(n, F_m)$ for $n$ sufficiently large
and proved that the extremal graph is unique, where $F_m$ is disjoint paths of
$P_{k_1}, \ldots, P_{k_m}$ [Lidick\'{y},B., Liu,H. and Palmer,C. (2013). On the
Tur\'{a}n number of forests. Electron. J. Combin. 20(2) Paper 62, 13 pp]. In
this paper, by mean of a different approach, we determine $ex(n, F_m)$ for all
integers $n$ with minor conditions, which extends their partial results.
Furthermore, we partly confirm the conjecture proposed by Bushaw and Kettle for
$ex(n, k\cdot P_l)$ [Bushaw,N. and Kttle,N. (2011) Tur\'{a}n numbers of
multiple paths and equibipartite forests. Combin. Probab. Comput. 20 837-853].
Moreover, we show that there exist two family graphs $F_m$ and $F_m^{\prime}$
such that $ex(n, F_m)=ex(n, F_m^{\prime})$ for all integers $n$, which is
related to an old problem of Erd\H{o}s and Simonovits.
