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So, imagine a world where magic is a real, quantifiable energy. It has real, visible effects, and can be used by humans to mold reality as they see fit (with varying degrees of finesse). The beginning for my story starts with my protagonist who is especially gifted in magic such that weird things manifest themselves without her even trying. She accidentally caused a little bit of a scare in her home town, and because of that, she is being sentenced to die. She escapes and starts on her journey to tame her power, reclaim the lost arts, and ultimately kickstart an age of magic.

Not terribly original, but the real problem is that I'm having a hard time believing that world with real magic could produce a civilization that's scared to death of it. That'd be like a civilization on Earth killing anyone who could run backwards, or start fires, or make tools. It's just a part of reality. You'd think humans would just use magic like they would any other tool at their disposal.

How can I justify a society that's fearful of witches in a world where magic is measurable?

Just make magic corrupting. Several books use the premise that the source of magic comes from other world beings which slowly cause madness.

The Wheel of Time series, the male wellspring is corrupted/poisoned so whenever men touch the source they slowly go mad.

Finally magic is power and power itself is corrupting. Even if magic doesn't actually affect you mentally or physically, suddenly you have a rare young person handed unlimited power and virtually nobody can stop them once they get the hang of it. It doesn't take too many blood thirsty conquering warlocks to make the peasants want to set fire to anyone that has the ability.

Just make it rare. Rare enough that most people wouldn't have direct experience with it. Then the situation wouldn't be any different from our own history when people thought that witches/magic was real even though they didn't ever actually see it.

And even if you can't make it rare... well, just look how fearful people can be of people of different skin colors, also not rare, and that doesn't even impart any potentially destructive abilities!

Basically you don't need a whole lot of justification to make people fearful irrational xenophobes (alas.)

The original reason we used to burn witches was competition. The church (insert denomination here) saw witchcraft as a threat. There is even some historical evidence to prove that the depiction of the devil as a red horned creature (as opposed to the beautiful angel of light he's described as in the Bible) was to make him look like Pan, a common nature god popular with certain forms of Wiccan belief. This was ultimately the first ever retrospective copyright case to convince the flock that witches worshipped the devil.

Even Nicola Tesla ran into persecution by offering to make electricity free to all.

Magic ultimately makes at least some forms of commercial industry obsolete and challenges the orthodoxy of belief that many are invested in. In such a case, burning witches is just another way to maintain the established power bases.

Violent people are part of society too. Arguably, an accepted part for thousands of years. Sometimes the capacity of an individual for violence is a tool which can help that person or his family or his society.

Why then would a violent man be sentenced to death?

Because he cannot control his violence.

Potentially dangerous abilities require control on the part of the individual. I think of Lennie in Of Mice and Men. A big huge simpleton - capable of great strength and immense amounts of work, but not in control of his powers and so doomed. I thought the same thing about Elsa the ice princess in Frozen - she realistically could have (would have) wiped out her country accidentally. Her lack of control meant she was too dangerous to let live. The same with the mutant Phoenix in Xmen.

Your society is not scared of magic. It is scared of an individual with the power to do great harm, and without the ability to control it.

Projection

Magic may be easily quantifiable in your world, like rain or sunshine, but people are still free to project their own cultural biases on it. All it takes is a handful of magic-users, or even a single individual to use their powers for less than honorable purposes, to create a lasting belief that Magic-User = Bad Person, therefore Magic = Bad. It doesn't even need to be something flashy like razing a farm, simple brigandry and charltanry would do.

Given the medieval mindset and the simple human desire for repetition, people born after this possibly apocryphal negative history with an aptitude for magic would be burnt at the stake or run out of town at the very least simply as a matter of standard village procedure to Ward Away The Bad Things. It doesn't matter what the new magic-user intends or whether s/he has done anything at all with magic, because the regular folk have been conditioned to believe that Magic = Bad