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What do we do if all the data we have access to really is horribly flawed?
Agnosticism is always an option
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In this case, you point to the flaws in the study and if better are available, cite those. If there’s nothing better, then provisionally accept what we actually have, and go from there. What you don’t get to do is simply say “study bad, therefore it’s all dismissed.” I’m still right because I’m rejecting the data I don’t like, and I reserve the right to reject any data I don’t like on the basis of whether or not I like the studies in question. It’s dishonest in a debate to give yourself the power to simply dismiss evidence without having some data of your own refuting it, Twitter surveys suck as evidence, but absent other evidence from better sources, you can’t simply say “bad methods, so it doesn’t count.” It refuted the point in question, that At least some women enjoy sex work. You can point out that you took a survey of people who follow a prostitute and therefore it’s biased, you can point to a lack of controls to prevent multiple accounts by the same person voting. It’s flawed, but it’s at least some evidence.
But why should I do this if, as posited, I have good reason to think that the data sucks?
Let me give a sort-of example from my own area of expertise. It's not actually a data-driven field; it's very deterministic mathematical theory. For decades now, people have been solving certain problems one way, using one method. The method has significant flaws. Some of the flaws are well-known; others, more damning ones in my mind, are just being revealed now. (I hate to say it, but it truly is, "Being revealed by a series of papers in which I'm a coauthor." I can lessen the arrogant-sounding sting a little bit by wholeheartedly acknowledging that it was a collaborator, not me, who came up with the initial counter-example that kicked off the whole shebang.)
We've been able to fix the problem, using a completely different method (established in a different context)... but so far, only for one specific version. There are numerous other variants of the problem. The thing is, for several of these variants that we've looked at, I can demonstrate that the (very bad) problem exists! I can show actual examples demonstrating why and how the prior methods fail to do what we had previously expected them to do. But we haven't yet 'fixed the glitch' for all these other variants (working on it!).
In sum, I know the bounds of what the prior method actually accomplishes, but I also now know what it doesn't accomplish. This has been hard for some people I've talked to in the field to grok, because they're so steeped in the old method. (I've had this conversation quite a few times, and it really breaks their brains at first, but if I get them to really focus on a particular example and I get them to really consider what would happen with the counterexample, I have a 100% rate of convincing them so far (profs in the field).) If someone were to say something like, "Yeah, ok, well, we know the prior method isn't perfect, but there's nothing better yet for this particular version of the problem, so let's provisionally accept it and go from there," I'm going to say, "HELLS NO!" Instead, I'm likely going to go find a particular counterexample for this variant, show exactly how the existing method is broken for this variant, and simply say, "We can't actually proceed further until we fix this."
I know this is shrouded in a small amount of mystery, but it's related, because we want to say, "Method/data says X." We think that, "Method/data says X." But it turns out that the method/data actually only says Y... which turns out to be very far from actually saying X. I'm not going to provisionally hold X when it pretty clearly says only Y and we don't actually have proper evidence for X.
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