The conventional model for splicing involves excision of each intron in one piece; we demonstrate this inaccurately describes splicing in many human genes. First, after switching on transcription of SAMD4A, a gene with a 134 kb-long first intron, splicing joins the 3' end of exon 1 to successive points within intron 1 well before the acceptor site at exon 2 is made. Second, genome-wide analysis shows that >60% of active genes yield products generated by such intermediate intron splicing. These products are present at ~15% the levels of primary transcripts, are encoded by conserved sequences similar to those found at canonical acceptors, and marked by distinctive structural and epigenetic features. Finally, using targeted genome editing, we demonstrate that inhibiting the formation of these splicing intermediates affects efficient exon–exon splicing. These findings greatly expand the functional and regulatory complexity of the human transcriptome.