In the present paper, to fill the gap of the effect of singularity arising from multiple fractional derivatives on numerical analysis, the regularity and high order difference scheme for time fractional telegraph equations are taken into consideration. Firstly, the analytic solution is obtained by employing Laplace transform, and its regularity is then deduced. Secondly, by the technic of decomposition, the improved regularity of solution is derived. Furthermore, to overcome the weak singularity and enhance convergence precision, a second-order fitted scheme based on L2-1_σ approximation and order reduction method is applied to such problems, which is an improvement for the work [6]. Ultimately, examples are presented to verify the effectiveness of our theoretical results.