The indication I refer to is that when you see a ship moving on screen from the perspective of another ship, it never suddenly blinks to "near lightspeed", because what that would look like is that in one frame the ship is there, and the next it isn't. That is never what happens. What happens is that the ship lumbers off, and slowly turns over the course of a couple of seconds, and can be seen to visibly accelerate in a way that the ship doesn't even cover its own length in less than five or six seconds. If you work the math it comes out on the order of 50-100G. Accelerating to near-lightspeed in a couple of seconds is several orders of magnitude larger than that.

Of course the ships don't blink around at near lightspeed; it wouldn't be photogenic. So all ships on all shows in all screens move within roughly a single order of magnitude of screen speed, no matter what the tech manuals say. Similarly, no matter what the claimed ranges of the weapons are, all engagements take place within a range where the two ships can clearly see each other, meaning no more than a handful of kilometers at most. No matter how far the ships are nominally apart from each other, a bolt of energy fired from one ship to another will take on the order of half a second to arrive. Almost no matter how zoomed in we are or are not, the energy bolts will subtend roughly the same amount of our eyes. If the ship is, say, firing on a planet, the bolts will perspective shrink maybe by a factor of two or three, if they shrink at all, rather than doing what they should do which is disappear entirely (except for perhaps one lit pixel if the bolt is sufficiently emissive). There's this one ST:TNG episode where the Enterprise uses phasers to quickly mine holes in the ground and when you start to see this sort of thing it's hard not to see the Enterprise as firing on a roughly 5-mile-diameter ball sitting right next to it, because if you follow the perspective of the "beam" that's what it implies about the size of the target.

Space isn't photogenic. Even today on earth, real combat takes place at ranges where you don't have a direct visual on the target a lot of the time. Any even vaguely realistic space combat will take place across ranges where you can't see a mile-sized ship with the naked eye. But that's not what they want to show. Consequently there's a massive, massive mismatch between the numeric claims of performance and what we actually see on the screen. So you might say something like "oh, the screen is maybe being symbolic", but then they routinely do things that imply what's on the screen is the real truth, such as in the one TNG movie where Data leaps between two ships. That's just one example, there's plenty of other situations like that.