Dual-function radar-communication (DFRC) systems, which can efficiently utilize the congested spectrum and costly hardware resources by employing one common waveform for both sensing and communication (S&C), have attracted increasing attention. While the orthogonal frequency division multiplexing (OFDM) technique has been widely adopted to support high-quality communications, it also has great potentials of improving radar sensing performance and providing flexible S&C. In this paper, we propose to jointly design the dual-functional transmit signals occupying several subcarriers to realize multi-user OFDM communications and detect one moving target in the presence of clutter. Meanwhile, the signals in other frequency subcarriers can be optimized in a similar way to perform other tasks. The transmit beamforming and receive filter are jointly optimized to maximize the radar output signal-to-interference-plus-noise ratio (SINR), while satisfying the communication SINR requirement and the power budget. An majorization minimization (MM) method based algorithm is developed to solve the resulting non-convex optimization problem. Numerical results reveal the significant wideband sensing gain brought by jointly designing the transmit signals in different subcarriers, and demonstrate the advantages of our proposed scheme and the effectiveness of the developed algorithm.