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.

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.

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 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.

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.