In my experience fast growing or slow growing they take a few years to establish, then your fast growing really cause a problem as they need constant maintenance.
A good evergreen hedge would be holly. As usual it is slow to start, but once established is fairly quick to grow and gives a nice looking evergreen hedge which is no problem to animals.
Don't forget that any young hedging plants will need protection from things that will eat them - for example, rabbits, sheep, cattle, horses etc etc. And also may need weeding out to prevent being overshadowed by weeds.
All the best
Sue