I am someone who frequents several imageboards. As often happens, because there's complete anonymity, people often say the most insane, outrageous and vile stuff imaginable. Against my better judgement I was arguing with one such individual who said something very very bad about acts with children (on the order of 'I such and such little girl's so and so'), copying his statements to reply in bulletpoint fashion. Eventually the mods stepped in and the argument was over, so I honestly forgot about it and went on with my day. Later, thinking I had a url copied, I instead pasted one of his very bad statements into my taskbar and searched Google with it. The very microsecond I realized what I did (that page hadn't even loaded fully), I closed the page and deleted my history, but I've been completely sick and horrified even having this associated with me, and I know such searches usually get the intention of the law. Could I legitimately face criminal prosecution over this? I have a completely clean criminal record and am not part of any ongoing investigation of any kind.
Could I legitimately face criminal prosecution over this?
It's not clear that you have committed a crime. If you're asking whether you might be suspected of having committed a crime, well, you have to identify a crime which you might be suspected of having committed. What would the elements of such a crime be?
You can have a look at, for example, New York's obscenity laws. It's difficult to see how a prosecutor could seek an indictment on the facts given in the question even without knowing that the words were not your own and that you had pasted them inadvertently into your search box.
If your worry about this is affecting you in your daily life then you should consider whether getting advice from a lawyer would be worthwhile, so you have a better understanding of the likelihood of prosecution and the likelihood of success of any defenses you might have available. It might cost a few hundred dollars, but if you are seriously troubled by the possibility of prosecution then the cost might be justified.