The sparse property of a direct adaptive equalizer (DAE) for single-carrier underwater acoustic communications is well recognized. It has been used to improve the performance and/or reduce the complexity of a DAE. Extensive investigations have been performed in terms of performance improvement. On the contrary, research on complexity reduction remains preliminary. A fundamental way for reducing the complexity of a DAE is to keep only significant taps while discarding trivial taps, that is, to run a partial-tap DAE. Existing partial-tap DAE designs assume a slowly varying sparse structure and may suffer performance degradation under a severe underwater environment. Motivated by this fact, the dynamic compressed sensing (DCS) technique is resorted to and a partial-tap DAE based on the sparse adaptive orthogonal matching pursuit-affine projection algorithm is proposed. The sparse adaptive orthogonal matching pursuit-affine projection algorithm-direct adaptive equalizer (SpAdOMP-APA-DAE) achieves symbol-wise updating of both positions and values of the significant coefficients. In this paper, a more extensive study on DCS-based DAEs is performed, and an enhanced dynamic compressed sensing-direct adaptive equalizer design enabled by the sparse adaptive subspace pursuit-improved proportionate affine projection algorithm (SpAdOMP-IPAPA) is proposed. The sparse adaptive subspace pursuit-improved proportionate affine projection algorithm-direct adaptive equalizer enjoys lower complexity while better performance than the previous SpAdOMP-APA-DAE. Experimental results corroborated the superiority of the SpAdOMP-IPAPA-DAE.