Impulsive noise occurs frequently in underwater acoustic (UA) channels and can significantly degrade the performance of UA orthogonal frequency-division multiplexing (OFDM) systems. In this paper, we propose two novel compressed sensing based algorithms for joint channel estimation and impulsive noise mitigation in UA OFDM systems. The first algorithm jointly estimates the channel impulse response and the impulsive noise by utilizing pilot subcarriers. The estimated impulsive noise is then converted to the time domain and removed from the received signals. We show that this algorithm reduces the system bit-error-rate through improved channel estimation and impulsive noise mitigation. In the second proposed algorithm, a joint estimation of the channel impulse response and the impulsive noise is performed by exploiting the initially detected data. Then, the estimated impulsive noise is removed from the received signals. The proposed algorithms are evaluated and compared with existing methods through numerical simulations and on real data collected during a UA communication experiment conducted in the estuary of the Swan River, WA, Australia, during December 2015. The results show that the proposed approaches consistently improve the accuracy of channel estimation and the performance of impulsive noise mitigation in UA OFDM communication systems.