By the large eddy simulation method, the turbulent Taylor-Couette flow of conducting fluid under a homogenous transverse magnetic field is investigated through using the computational fluid dynamic software ANSYS Fluent 17.0. The flow is confined between two infinitely long cylinders, thus a periodic boundary condition is imposed in the axial direction. The inner cylinder rotates while the outer one is at rest, and their radius ratio is 1/2. Two Reynolds numbers of 3000 and 5000 are considered in the simulations, and the Hartmann number is varied from 0 to 50. In the present study, we assume a lower magnetic Reynolds number $Re_{\rm m} \ll 1$ , i.e., the influence of the induced magnetic field on the flow is negligible in comparison with the imposed magnetic field. The evolution of Taylor vortices, velocity profile of mean flow, and turbulent kinetic energy distribution under the transverse magnetic field are analyzed and compared with the results of the axial magnetic field counterpart. It shows that the imposed magnetic field has a significant damping effect on the Taylor-Couette flow. The twisted Taylor vortices break into small-scale vortex structures under the transverse magnetic field and they arrange themselves along the magnetic field. The fluctuations which are perpendicular to the magnetic field are suppressed effectively, while the one which is parallel to the magnetic field is nearly uninfluenced, resulting in quasi-two-dimensional elongated structure in the flow field. As anticipated, in a sufficiently strong magnetic field, the turbulent Taylor-Couette flow may eventually decay to a Couette laminar flow. In the outer cylinder and the area perpendicular to the direction of the magnetic field, the suppression effect is even stronger than those in any other places and fewer vortices are observed in the simulations. The turbulent kinetic energy is transferred firstly from large eddies to intermediate eddies, then to small eddies, and finally dissipated due to the viscous and Joule effect. As the Reynolds number increases, the suppression effect of the magnetic field weakens, and the flow behaves divergently in different areas of the apparatus. Compared with the axial magnetic field, the transverse magnetic field has a weak suppression effect on the flow field, and the profiles of related variables are obviously anisotropic.