We present improved characteristics of the curved plasmonic waveguide which consists of a thin metal stripe with asymmetric cladding layers. It is shown that in the proposed curved asymmetric plasmonic waveguides, a balance between a radiation due to bending and a radiation due to the asymmetric claddings allows a bending with a smaller radius curvature and a lower loss compared to the waveguide with symmetric claddings. At the same time, a symmetric metal stripe waveguide's typical trade-off between the bending characteristics and the propagation loss of a straight waveguide is relaxed with proper amount of asymmetry. With the proposed structure, a plasmonic waveguide bending whose radius is as small as 2 mm with a total loss of 1.8 dB/90 degrees is designed. Enhanced sensitivity to the surrounding medium and its application are discussed.