There's nothing stopping you from starting the immigration process in the US if you entered legally on a valid visa.

The wrinkle comes with the type of visa, either non-immigrant intent, immigrant intent, or dual-intent.

On a non-immigrant-intent visa, like a student visa or a tourist visa, if you somehow happen to qualify to apply for a GC (say, via a prospective employer) you still can start the process in the US even though you're in there on a non-immigrant-intent visa. It's just that if you leave and try to re-enter, you may be refused entry while your application is processing. If you have an immigrant or dual-intent visa, this is not an issue.

For people who's visa status expires in the US while they're waiting in line, they have to leave the US immediately until their GC is processed, with a few exceptions. Those who have an approved GC petition (having waited in line for however long until a GC became available, but after that point it takes another 4-6 months for processing) can apply for "adjustment of status" and "advanced parole". This gives them legal status in the US, and allows them to be "paroled" back into the US on a provisional basis (but they can still be refused entry, it's just unlikely) until their greencard is actually issued.

And in certain very specific cases, such as for H1B visa holders, even if their greencard has not yet become available (e.g. Indian nationals have a wait time of 8-10+ years until a greencard becomes available for them) they are able to extend their work visas indefinitely until their greencard petition is adjudicated.