I think that you have a couple of points wrong. Since your question is asked using bacterial terminology, I'll stick to that.
The leading strand, the one that is initiated at the origin of replication, is synthesised by pol III which is a highly processive polymerase, i.e. it keeps on going for long periods, making very long products. In principle a single pol III molecule could produce the entire leading strand copy of the template.
The lagging strand is, as you say, initiated at multiple RNA primers. The enzyme that extends these primers is pol I. As polI extends a primer it creates an Okazaki fragment (i.e. this is not an alternative name for the primer itself). Eventually the pol I will encounter the 5' end of another RNA primer. At this point the 5'>3' exonuclease activity of the pol I comes into play and removes the primer, replacing it with DNA. The pol I will probably also degrade some of the DNA that has been added to that primer by another pol I, resynthesising it as it goes. This is so-called "nick tranlation" since as the pol I moves along the template it moves a nick in the new strand as it goes. pol I is not a very processive enzyme however and will fall off, leaving the nick to be sealed by DNA ligase.
The leading strand does require a primer, and in most cases this is an RNA laid down by the initiation complex at the origin of replication. In some cases, a protein provides an -OH group for DNA polymerase to initiate at.