Emerging maritime applications arising from the continued growth of the marine economy have an inherent need for high data rate underwater wireless links. Within this context, underwater wireless optical communication is known as a promising technology for data transmission over short-to-medium ranges; the current available technology provides a transmission span of about 100m. In view of extending the transmission range, silicon photomultipliers (SiPMs) have recently emerged as a photodetection solution offering high receiver (Rx) sensitivity together with operational flexibility. In this paper, we introduce the use of pulse-amplitude modulation (PAM) together with frequency-domain equalization (FDE) at the Rx to boost the communication rate beyond the bandwidth (BW) limitation of the optoelectronic components. For instance, for a link BW limited to 2MHz and 2-PAM transmission with a target bit-error rate (BER) of $10^{-4}$, the link becomes nonoperational for data rates higher than $\sim\! \text{8}$Mbps without equalization, whereas much higher data rates can be attained using FDE, e.g., 20 and 50Mbps with maximum ranges of 28 and 10m, respectively, in clear waters for the SensL MicroSB-30020 SiPM and an average transmit optical power of 0.6W only. Meanwhile, the nonlinear distortion of the SiPM is shown to limit the modulation order and thus the data rate in relatively short ranges. We also propose appropriate processing for PAM modulation and demodulation, given the quantum-noise-limited Rx when using an SiPM. We show that the use of nonbinary PAM is undeniably advantageous for moderate data rates (symbol rate a few MHz higher than the overall link BW) when no channel equalization is performed at the Rx. However, when employing FDE, only for very high data rates (e.g., symbol rate ten times higher than the link BW), where the link frequency response becomes highly frequency selective, the nonbinary PAM becomes practically interesting, outperforming 2-PAM.