Red seaweeds form part of an ancient lineage of eukaryotes that were one of the first to evolve multicellularity. Although red seaweeds share a common evolutionary origin with modern-day plants and display complex multicellular development, we still lack comprehensive genome data from the most highly-evolved groups. Here, we present a chromosome-level genome assembly of the complex red seaweed Bostrychia moritziana, a member of the largest and most diverse order of red algae called the Ceramiales. Contrary to the commonly held view that red algae generally have small genomes, we report significant genome size expansion in Bostrychia and other Ceramiales species, which we posit as one of at least three independent genome expansion events that occurred during red algal evolution. Our analyses suggest that these expansions do not involve polyploidy or ancient whole genome duplications, but in the case of Bostrychia appear to be largely driven by the dramatic proliferation of a single lineage of giant Plavaka DNA transposons. Consistent with increased genome size, we identify a substantial increase in gene content in Bostrychia that was shaped both by de novo gene emergence and by the amplification of gene families in common with other Ceramiales seaweeds, providing key insight into the genetic adaptations underpinning the evolutionary success of this species-rich order. Finally, our sex-specific assemblies enabled us to resolve the UV sex chromosomes in Bostrychia, which feature expanded gene-rich sex-linked regions. Notably, these sex-linked regions each harbour a distinct TALE-HD transcription factor orthologous to ancient regulators of haploid-diploid transitions in other multicellular lineages. Together, our findings offer unique perspectives into the genomic adaptations driving red algal diversity and demonstrate how this highly successful group of red seaweeds can provide insight into the evolutionary origins and common principles of complex multicellular plant life.