Hi Roseanna, and welcome to ATP
You should probably start with a so called "Climbing" type rose, to be successful with this. Google "climbing rose pictures" to see lots of different applications including some very similar to what you're describing.
They don't really climb, but make long flexible canes instead of stiff stems like regular roses have. As they grow, you would just tie the canes along the top of the fence, or lower down if you want two 'tiers' of canes. But keep in mind that the lower canes need to get enough sun to bloom so should not be shaded by the upper canes. When they're tied horizontally, they will put up blooms all along the canes and be really gorgeous. There are lots of nice cultivars of climbing roses available.
If you already have some regular roses planted, you could just keep them pruned down to the height that is tolerable to you, and 4ft. is not an unreasonable height for regular rose bushes. But most really won't take too kindly to being 'trained' horizontally. They naturally grow upwards.