Due to its high spectral efficiency and robustness in multipath channels, orthogonal frequency division multiplexing (OFDM) has been considered one of the most promising coherent underwater acoustic (UWA) communication technologies. Channel estimation (CE) plays an important role in OFDM receivers for mitigating the negative impact of UWA channels, unfortunately, the performance of the conventional CE algorithms suffers from significant degradation under time-varying UWA channels due to the unavoidable mismatch between the CE and OFDM demodulation. As channel estimators are generally designed to optimize the signal-level cost functions. However, the results of which are used for bit-level OFDM demodulation. In this article, a minimum bit-error-ratio (BER) sparsity exploitation (MBSE) CE algorithm is proposed under the least-squares framework to address this mismatch. By tuning the sparsity exploitation parameters, i.e., threshold and diffuseness, in a novel manner, the proposed algorithm is designed to adapt to the time-varying UWA channels to improve OFDM demodulation. Specifically, while the sparsity exploitation threshold is obtained via minimum BER searching to determine the dominant multipath arrivals, the structural sparsity parameter, defined as multipath diffuseness, is symbol-wise updated to track the channel variations between the adjacent symbols. Numerical simulation and sea trial experiments verify the performance enhancement of the proposed algorithm in terms of output signal-to-noise ratio, CE mean-square error, and BER under artificial and physical time-varying UWA channels, respectively.