(all the pictures in the slideshow are of tower bridge too)
9 points
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Londoner here. The easy way to remember it is, the bridge with the towers, specifically the one next to the tower of london, is called tower bridge.
The one that was the only bridge into the old, walled city of London for centuries, specifically the one right next to London bridge Station, is called London Bridge.
Being used to it, I can only imagine how confusing that must be for people.