Dna replication occurs at the same time in both strands of dna. why aren't both strands extended continuously (i. e. why is there a leading and lagging strand)?

Because on the lagging strand DNA replication occurs in opposite direction to the unwinding of the double helix. That is it unwinds 5’-3’ direction but DNA polymerase reads the template 3’-5’ to synthesize 5’-3’. As a result it has to wait a little for enough nucleotides to be available to recreate a primer and elongate. This creates Okazaki Fragments.

This is the most closely resembles the model.