The quality of underwater imaging is greatly impacted by the scattering and absorption of light in turbid water environments. Single-pixel imaging (SPI) has emerged as a promising solution for turbid underwater imaging, as it effectively suppresses the effects of scattering and is cost-effective due to the use of a single photodetector. However, the quality of SPI in highly turbid water is still unsatisfactory. To address this issue, we propose a novel computational framework for turbid water SPI. The framework involves a machine-learning-based polynomial regression fitting method, followed by data feature enhancement in the spectrum domain to obtain the rectified data, and ultimately, high-contrast image recovery. Furthermore, we propose a new metric, edge-detection-based enhancement measure evaluation (EDEME), to quantitatively evaluate the contrast of the recovered images. Our experimental results demonstrate that our proposed method can recover images in low turbidity water to a level comparable to clear water, and even in highly turbid water (turbidity greater than 50 NTU), the recovered images are legible with significantly improved EDEME values. In addition, our method exhibits wide adaptability, requires minimal data operations, and outperforms some post-image processing methods. This work has significant implications for imaging, inspections, search and rescue, resource exploitation, and other applications in underwater environments.