Chiral magnetic materials provide a number of challenging issues such as the highly efficient domain wall (DW) and skyrmion motions driven by electric current, as of the operation principles of emerging spintronic devices. The DWs in the chiral materials exhibit asymmetric DW speed variation under application of in plane magnetic field. Here, we show that such DW speed asymmetry causes the DW tilting during the motion along wire structure. It has been known that the DW tilting can be induced by the direct Zeeman interaction of the DW magnetization under application of in plane magnetic field. However, our experimental observations manifests that there exists another dominant process with the DW speed asymmetry caused by either the Dzyaloshinskii Moriya interaction (DMI) or the chirality dependent DW speed variation. A theoretical model based on the DW geometry reveals that the DW tilting is initiated by the DW pinning at wire edges and then, the direction of the DW tilting is determined by the DW speed asymmetry, as confirmed by a numerical simulation. The present observation reveals the decisive role of the DW pinning with the DW speed asymmetry, which determines the DW geometry and consequently, the dynamics.