There's a reason for every weird thing people naturally do, and to truly work around the weirdness you need to understand the reasons and find them compelling.
People do weird things all the time. Eventually someone notices and gives it a name. For instance, whataboutism.
Thinking about it superficially, the term immediately explains itself. It's a fallacy. A thing your opponent does wrong, because their thought is broken.
If you're smart, you have some logical-sounding justifications for why whataboutism is wrong. They would normally move you much less if not for your victory in this argument depending on them, but they're not wrong.
(If you're dumb, you also have justifications, just they're dumb. Maybe it's even "everyone knows it's a fallacy, it even has a Wikipedia page".)
But for smart and dumb alike, functionally this is a score-on-your-opponent-free card. Just like the what-about card itself.
– I don't like this and that.
– What-about-card.
– Whataboutism-card.
Whoever plays the what-about card usually doesn't accept the version of the rules that added whataboutism-like-cards, so they just shrug it off.
Is this an improvement in awareness on your part? I dunnooooo. You have a point, but you also completely missed the point your opponent was making.
You're smart! How could this have happened to you?!
Your point is that this is focused discussion, not shit-throwing. It's cheap to counter claims with "but what about my claims". If the original claim is true then it's true and must be admitted!
Their point is they think you're lying and this is shit-throwing.
This is a totally reasonable assumption because it's usually true.
You failed to address this point at all.
So they're making a very good point, and succinctly move to respond in kind, and all you do is whine "wait wait, we must stay in my illusions that I'm reasonable and totally not partisan".
And now you even have a card to deal out your pretense on a reflexive basis, without even a chance to consider what your opponent is saying about this discussion, and how you really need to act if you really truly want this to be not a shit-throwing.
So what would be a good response in this case?
There's not always one. Often, if you reflect fairly, shit-throwing was your entire goal.
But for one, try to dissociate from the other side. Throw some fair shit that way too. Give some fair praise for the side you're attacking. Explain how this is your side, which is why you're much more interested in fixing this, than whatever you-toos also did. You can't do anything about you-toos, but this here is something you can make your own lives better about.