Plant a border of assorted shrubs all (or most) of the way along the fence, instead of having something stiff looking just in that one place. As for fast + tight that is a rare combination because in general it is slower growth that results in a more compact habit.

In my area the go to plant for narrow evergreen screening is Thuja occidentalis 'Smaragd'. This doesn't shoot up rapidly but it can be found fairly easily already head high. Including at warehouse stores here, in season.

your basic problem is you hope for a quick solution ... that isnt going to happen rowing trees or shrubs ....

i quickly solved a similar issue with window films .. extremely easy to apply with soapy water and a squigee ... so the trees i planted could a decade to row to height ...

https://duckduckgo.com/?t=ffcm&q=window+films&iax=images&ia=images

i ran across this site.. cruising the web ... not cheap.. but would solve the issue rather quickly.. what we call.. checkbook gardening .. and nothing wrong with that ...

but if that isnt in the cards .. look to the plants they suggest to use

https://www.instanthedge.com/hedge-varieties/

ken

"... and even with the fence my bathroom looks directly into his bedroom." So what's the problem?? The planets are lining up for you and you're sounding ungrateful. :-(

A strategically placed large shrub, in the tree form, will block the view and not take up too much yard space.

Whatever you plant, the first thing I would do is frost the bathroom window and put lattice in front of it with an evergreen vine and out a blackout shade inside. I would, if it was my house, take the window out completely! We have two bathrooms and neither has a window. They do have a great exhaust fan. IMO, there should never be a window in a bathroom for any reason.