In ST:TOS, we see stars slowly come from the front and move to the side as the crew looks forward in normal flight. Even this makes it look like their speed is at least several light years per minute. In some last shows, we see stars pass by side windows even faster. But the official conversion of standard warp speeds generally measures these in light years per hour. Obviously the effects people just wanted something we could immediately notice in a few seconds to indicate the ship's speed, but I wonder if there's any retcon or fanwank explanation for this. Like, if we're not actually seeing stars but some other astronomical features going past, which are closer together than stars are.
In my answer to this question:
I suggested that the "stars" seen streaking by are not stars, but much smaller and closer objects, which may be glowing as a result of interacting with the warp field of the starship.