My dad doesn't want me to touch alcohol before I turn 21.
Is it perfectly natural to use 'touch' here? I searched for it and found some examples, but not enough to make me sure.
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It is a form of clichéd hyperbole, but so natural and common that it may not be noticed as such. What the father literally wants is for the child not to drink alcohol. Drinking generally requires touching, but it is possible to touch without drinking. (The same father may sometimes tell the child, "Bring me a can of beer," and not think himself inconsistent.) The point is to make the prohibition exaggerated and emphatic -- don't even touch it (let alone drink it). A more extreme version would be, "Don't touch it with a ten-foot pole."