Prodogg asked: How would Azula and Zuko try to convince everyone that their relationship is ok?
The only ones they'd truly be trying to convince would be the people they love (friends and family), and they'd do that just by being shamelessly enjoying being a ridiculously loving, happy couple. It take a while, but eventually they'd all at least be willing to ignore their own feelings about incest to support them, or even start thinking "You know what, incest isn't so bad after all"
As for everybody else, it'd depend on what group they'd fall into (since this is about convincing people to tolerate/not get in the way of their relationship, I'll write these assuming that Zuko and Azula's family was the ultra-rare type of royalty, aka the ones that weren't, at very least, screwing a cousin)
Fire Nation Nobles
When it came to the ones that were just worried Azula would convince Zuko to start a new war or even kill him once she no longer needed him, they'd demonstrate through direct action that this was not the case, with Azula working really hard to help Zuko (and by consequence the Avatar) keep balance. As for the more "ambitious" nobles, who are just bitter that they lost their chance of becoming royalty, they'd help them find good matches for marriages that would benefit them in exchange for support. In both cases, a subtle reminder of "I hold more political power than you, I have an army, I'm stronger than you, and I'm friends with other people who are powerful than you" would help out if anyone gets too confrontational.
Regular Fire Nation Citizens
They'd win them over by being good rulers - the best rulers, in fact. There would always be people who would never accept incest, but if human history has taught us anything is that the vast majority of people will put up with anything if it means they're safe, have a roof over their heads and food on their table.
Regular Foreign Citizens
They wouldn't really have to do anything to convince them, and their support isn't even that important. They'd be afraid that Azula would influence Zuko to start a new war, but once it was clear that this wasn't going to happen, they'd just chill, even if they are disgusted by it - it's not like Zuko and Azula are their rulers, so it's not their problem.
Foreing Nobles/Royals
To deal with the ones that were just personally disgusted by incest, they'd use good old diplomacy to show that this just isn't a battle worth fighting since the issue does not affect them in any way, and they need to focus on rebuilding their own nations/cities (which Zuko and Azula will help them do, mostly with some money). As for the nobles that are just using it as an excuse to try and get revenge against the Fire Nation (and the ones that are using revenge/justice as an excuse to try and become conquerors) they'd casually remind them that they don't want a war... but that they will fight if they have to, and that they will win.