I assume that is your fence so why don't you think about growing vines or espaliered plants. They will take up less room but give you a sense of privacy and then plant perennials or spreading woody plants on the border.
Pyracatha would look great in the winter as long as you don't have children, clematis and some evergreens and how about a jap maple.

If you mean that the utilities run 2 feet away from the fence, you can probably plant a small tree or two right off the end of your patio and limb it up as it grows. It will give you privacy when you're using that area, at least.

If you need screening while you're using other parts of the backyard, some tall shrubs planted far enough away from the fence to avoid interfering with the utilities will add some privacy.

If you are referring to being able to see through the cracks between the boards in the fence, well, you could nail or screw battens over the cracks or put a second set of boards on your side, or grow vines on the fence. Do your neighbors actually put their eyes to the cracks to look at you, or are you doing something naughty back there that you don't want to be seen doing? LOL

I live in a duplex with no fence so I understand the need for privacy. Vines are a great idea or how about grasses. Where exactly is underground utilities and gas pipe. I had the utility come out and paint where the gas line is. From there I put out shrubs and grasses that don't interfer. I even planted several tree but height is more of a problem for me, I have overhead wires so a little research was need.