My garden fence is very similar to yours--a white picket fence. And, as others have said, the birds would fly to the top, land there, and then hop down rather than truly flying over the fence. However, after observing them I noticed that they only landed on the fence posts, not the pickets. My pickets are pointed at the top and spaced about 4 inches apart. Somehow this was too difficult for them to balance on.
So, you could consider putting something on top of the fence posts only that'd make landing on them impossible. You might find something attractive like a giant metal or wooden flower that would look nice but be too spiky or slippery to land on. Or a bird house or a weather vane or a rain gauge....anything that would prevent landing/perching.
I was too impatient to find something like this so in the end I nailed a 12 inch extension to the top of each post made from 2x3 strips of wood. I drilled two holes in each extension and ran 17 gauge wire so it sits 4 inches and 7 inches above the top of the picket fence. It's not real pretty but it does the job. The purpose of this is that they hit the wire when attempting to land on the fence post top but the wire is too hard to see and too flimsy to hold their weight so they don't land on it (I purposely did not tighten the wire as tight as it could go).
By the way...clipping their wings didn't work as my New Hampshire Red could still hop about 4 feet to the top of the fence post. Might work for other birds, though.