You are correct in that there is a constant flow of water down the river. However that water eventually meets the sea, and in ELI5 terms: there is a lot more water in the sea than in the river!
So when the sea level rises due to the tide, it literally pushes the water back up the river and the river’s level rises too.
As others have mentioned, in some rivers the geology allows you to actually see the water being forced back upstream in the form of a tidal bore. In the UK there is sometimes a strong tidal bore on the Severn — strong enough to surf on! — and I’ve seen it myself on the Mersey where it narrows around Runcorn. Also if you go to Conwy in Wales, you can stand on the bridge by the castle and watch the river flowing inland away from the sea when the tide is coming in. I’m sure there are many other examples in the UK and abroad.
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