Make a list of the features YOUR site is not offering and you think you desperately need, then find the software that can give you exactly that. Otherwise you get what somebody else thinks you need and you'll forever fight to get it to do what you want, no matter how much it might improve the existing setup.
If image support LINKED to a forum structure is what you need more than anything else, look to softwares that will allow you to "plugin" image server features that you have 100% control over.... either as a local server or cloud (or single purpose host solution). This for PERFORMANCE reasons. (This part is not cheap) To really control things have your own server/rack/cloud and don't piggy back off anyone else.
If dealing with kiddies, have software that will age verify AND limit access to areas for adults only.
You mentioned governments interested in what you are doing. Software can't deal with that, but clear cut policies, moderation, and never letting up on either is your best bet .... though having an attorney or two in reserve would be nice.
You can't fake moderation with software. G tries (see youtube for how well that is going) and ultimately it is human eyeballs (ask Facebook, or any major newspaper that has NOT turned off their UGC commenting) that gets the job done. Software won't fix that, but being able to apply FILTERS to expose problematic messaging can help, so whatever software you look at must be open enough to allow customization to create (IN ADDITION TO WHATEVER THE SOFTWARE CREATOR THOUGHT YOU'D NEED) any special criteria that will come up in operations.
Expect to spend money. Might even have to hire an "editorial moderation staff" (as in more than one) at salary and make THEM responsible for what's got you buried (in over my head as stated) at the moment.
Or... age verify, set a minimum age to join (adult, no kids!), clean up your registration (that part should REALLY be automated, no email hand holding required!), establish a budget for your data storage on infrastructure robust enough to handle traffic and bandwidth, and go from there.
There are no magic answers.
The softwares suggested above have a lot going for them, but the exterior problems you addressed in the OP go way beyond what software alone can do.
Ultimately, whatever you do, if images are your basic method of operations, expect costs to increase ginormously as both bandwidth AND THE GROWING STORAGE PROBLEM will not go away and will only get worse.
What you really need is a Venture Capitalist. Have you considered putting together a proposal and hitting the bricks?
All the above it merely a reality check.
What I DID hear is that moderation and registration are the two things getting out of hand.
Fixing the registration process should be step one and that has nothing to do with the software in use at the moment, though whatever you do will have to pass that info on to the forum software to actually enroll new members.
The second aspect, moderation, is a bit more difficult, but not impossible. What kinds of moderation are required? Text? Copyright image control? Personalities running wild?
Knowing how to code websites has nothing to do with moderation of content. For that you need maturity, commonsense, verbal and interpersonal skills, and a joy at "fixing things".
Look for a retired school teacher in your area. Or police officer, sports coach, doctors or nurses, propose a reasonable work schedule and compensation (which can be trade, cash, or other considerations) and go from there.
I'd like to say there's a piece of software out there that will do exactly what you want for the price you're willing to pay, but such a thing does NOT exist.
If it did,. I'd be using it and I wouldn't tell you!