Needle insertion procedures are commonly used for surgical interventions. In this paper, we develop a three-dimensional (3D) closed-loop control algorithm to robotically steer flexible needles with an asymmetric tip towards a target in a soft-tissue phantom. Twelve Fiber Bragg Grating (FBG) sensors are embedded on the needle shaft. FBG sensors measure the strain applied on the needle during insertion. A method is developed to reconstruct the needle shape using the strain data obtained from the FBG sensors. Four experimental cases are conducted to validate the reconstruction method (single-bend, double-bend, 3D double-bend and drilling insertions). In the experiments, the needle is inserted 120 mm into a soft-tissue phantom. Camera images are used as a reference for the reconstruction experiments. The results show that the mean needle tip accuracy of the reconstruction method is 1.8 mm. The reconstructed needle shape is used as feedback for the steering algorithm. The steering algorithm estimates the region that the needle can reach during insertion, and controls the needle to keep the target in this region. Steering experiments are performed for 110 mm insertion, and the mean targeting accuracy is 1.3 mm. The results demonstrate the capability of using FBG sensors to robotically steer needles.