It's normal to not want to fail. To actively fear failure is not the same thing.
No one likes to try and then not succeed but we all realize it is part of the learning curve. You can't hop on a bike and balance on the first try. You can't ice skate without practice. You can't run a large, complex project without falling off the rails a few times along the way. (same goes for parenting)
Yet there is a growing sense in our careers (and sometimes personal lives) that we must avoid failure at all cost." Ick. Mistakes will happen; try to not make the same one twice.
This avoidance turns into a fear of failure that is different from doing one's best to ensure few errors happen through good planning and open conversation. Fearing failure means that instead of taking measured risks and knowing there is a tolerated margin of error, one defers decisions or refuses all risk resulting in a poorer, more limited outcome.
Fear is not always a bad thing. Making driving choices because mistakes could be fatal is a good idea. In other, less life/death instances, ask "what's the worst thing that could happen?" Chances are you could deal with it, plan for it or… perhaps it is just too outrageous/irrational a fear to have a real chance of occurring. If everyone got fired or unloved for every mistake, there's be lots more unemployed lonely folks out there…