In a video on youtube, it says "we prefer 'over' to talk about a movement to the other side of sth high. We prefer 'across' to talk about a movement to the other side of a flat area".
I think "a bridge across the river" means the bridge is a flat bridge that is nearly above the river, while "a bridge over the river" means the bridge is an arch bridge. Am I right?
When talking about the physical structure of the bridge itself, in general you would be correct. "Over" doesn't necessarily mean only an arch bridge — any bridge that is substantially above, such as a suspension bridge, could be implied.
However, one could also emphasize not the physical bridge but the passage or motion. In this case, you could say "across" whenever the emphasis was on the crossing (getting from one side to the other) and "over" whenever the emphasis was on the state of being above/distant from the river. "He needed a bridge to get across the river" means that he is focussed on being on the other side of the river, and it doesn't matter what kind of bridge he ends up using. "Like a bridge over troubled waters" emphasizes that the bridge is removing you from the dangerous situation.
Google Ngram shows that "a bridge across/over the river" are variants of almost the same frequency. As already said, the two prepositions don't say anything about the kind of bridge.
A bridge over/across the river
I think you can use either over or across. Both the prepositions convey the same meaning i.e. from one side to the other side of the river. Please look at the following sentence from Cambridge Dictionary:
We drove over/across the bridge.
The video on YouTube says that we prefer to use over in this case, but it doesn't mean that the use of across is wrong. So when you are talking about a movement from one side to the other side of something high, you can use of the prepositions, though "over" is preferable. It doesn't matter whether the bridge is a flat bridge or arch bridge.
The purpose of building a bridge is to cross the river; so "across" is used in the context of traveling or moving. E.g., Robison drove on the bridge across the river.
If movement is not to be emphasized, then "bridge over the river" is correct: e.g., A bridge has been built over the river.
The bridge is at a higher position and there is no contact between the river and the bridge. So, ‘over‘ can be used:
There is a bridge over the river.
Again, the bridge crosses the river from one side to the other. So, ‘across‘ can also be used :
There is a bridge across the river.
Across doesn't mean above, it may mean bridge has touch with the water but over shows bridge has no contact with water.
Think the best preposition that goes with it is ACROSS. Over suits, but think the one that matches best is across. So it's absolutely no doubt that the bridge is built across the river. Very objective.
