川 and 橋: a watercourse and a structure spanning a gap
Compare a river with a bridge over it, using a bridge above railway tracks to avoid assuming that every bridge crosses a river.
A river or stream
川, read かわ, means river or stream. この川は町の中を流れています。 says that this river flows through the town.
川の幅を測りました。 says that someone measured the river’s width. Unlike the existence statement 川がある, 測りました explicitly reports a measuring action.
A structure connecting across something
橋, read はし, means a bridge spanning something such as a river or road. 川に橋が架かっています。 says that a bridge spans the river.
線路の上に橋があります。 says that there is a bridge above the railway tracks. What lies below a bridge is not necessarily a river.
Dimensions of a watercourse and a structure
橋の長さと川の幅は同じとは限りません。 says that the bridge’s length and the river’s width are not necessarily equal. It compares dimensions of different objects.
橋の下を船が通りました。 says that a boat passed beneath the bridge. It describes the boat’s route without stating that a person walked across the bridge.
Does 橋があります。 establish that a river lies below it?
Show the answer and why
No. A bridge can also cross railway tracks or a road. What it spans needs separate information.