Real quick question [duplicate]

It's not wrong, but it's informal.

The Free Merriam-Webster Dictionary has this to say:

Most handbooks consider the adverb real to be informal and more suitable to speech than writing. Our evidence shows these observations to be true in the main, but real is becoming more common in writing of an informal, conversational style. It is used as an intensifier only and is not interchangeable with really except in that use.

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As ruakh wrote, it's informal.

Is my previous sentence wrong? Must the real part be "really"?

This seems to be from American English, so the answer will depend on use. I have heard Americans use the word real like this. It would not necessarily be wrong for someone speaking American English. The word should be "really" for people speaking the British kind of English.

Which version to use will depend on which English you speak.

No it's not wrong. It's an informal form of the adverb really and very used as a submodifier.

My tooth hurts real bad,

My tooth hurts really bad, and

My tooth hurts very bad

all mean the same thing.