I'm going to push back on the people who try and shut this conversation down all the time and ask why does it bother you(follow your own advice and ignore the conversation) if some City fans do want to talk about the facts and beat them at their own game? That's essentially what it is, a game to these trolls... "What time frame, what stat can I cherry pick and/or take out of context to put City in the mud and seem small". You'll see it's exactly what they are doing, when you remove the feigned objectivity(usually comes before saying something they know is designed to inflame). I can see where the OP has been misled into believing some of this shite, so maybe we should be discussing it and setting the record straight.
I feel like, where the average PL fans(and even pundits and youtube presenters) get their pecking order criteria is completely inconsistent at best, or plain arbitrary in a lot a of cases.
Villa and Everton can point to their histories and success and have a conversation with Liverpool and United fans about that. So we have to give them a nod. They are well supported too but none more so than City.
But Leeds, Newcastle, Spurs, Forest etc never really were bigger or had more history than City. That's always been a bullshit narrative. History, support, success, longevity. I will get the fact that City have more topflight seasons than any of them out of the way to go in the longevity column.
Spurs - This has been discussed a few times on here, they never were bigger or smaller than City. They only started pulling away away in success, in the era where City started going down hill, which was the 70s. They still only had 2 more than City by the end of the 70s though, which isn't a lot. From what I've read, money had a lot to do with that too("they bought it" ;) ). As for history, City arrived as a big club before them, in the same era Arsenal had their first successful era(30s). Of course they had a better 90s but that wasn't hard. It was City's worst era, it's bit disingenuous to compare a normal era for one club to the worst of another club but people do that a lot I've noticed. City have pulled away from them since the takeover in every measurable way and they have to accept it.
Newcastle - They are a one club city in the North East, so being well supported should be a given. They are competing with no-one for a big population and it's been that way for their whole existence. Compare the attendance data properly though and there was never anywhere near the difference some people in the media(Chris Sutton, Simon Jordan type bellends who think they know what they are talking about but don't) have tried to say there is/was in terms of support. The 50s(their last success era) and the 00s(mostly before the move to the Etihad) at a glance seems to be as big as it got and there were reasons for that such as capacity of the stadium(City have a higher attendance since the expansion) and the club performances as the main reasons for it. As many City fans know, City maintained a bigger following like for like when both teams took the drop, so they can't really say a thing about City's support. In terms of success, they only had one more major honour than City before the takeover and the majority of the big ones came when football was still in it's infancy. I've seen some of their fans trying to list the Inter-Cities Fairs Cup(1969) and even the Inter-Toto Cup(2006) as a major honours when UEFA themselves don't(it was their competition), which is just unnecessary IMO.
Leeds - Again, well supported but that should be a given as one club city with such a population. Compare the attendance data and like Newcastle, it's a myth that they ever really had a larger support overall. They were late comers to the big club table compared to most of the names mentioned and they've never been in the top 10 most successful english clubs(5 total) at any point as a result of that. City were already two time champions of England before their first title.
Forest - Not really ever had the same level of support as any of the above and therefore always had a smaller following than City(not knocking them for that, still a fairly respectable following all the same). As others have said, they only really had one successful era(an impressive one though) under a great manager. They were even later comers to the big club table than Leeds(late 70s). City had already had two successful eras before then. They were still only level with City on major honours before the takeover. I don't feel like Forrest fans are the ones making noise about City in the same way some Leeds and Newcastle fans try to do, in the name of "bants".
I decided against including seasons in the topflight for each club but I did notice both Burnley(59) and Forest(56) have more topflight seasons in total than Leeds(52) at present. This might all be boring to some but the point is, personally, I think it's pretty mad considering some of the teams that some people try and place above City in the pecking order, when you lay out the facts.
Before anyone points it out, I realise the conversation is "are City a big club now". However, the OP mentioned clubs like Leeds and Forest as examples of big clubs and we've all seen people reason why City aren't a big club in that way. So the closing point is, even before the takeover, none of those clubs were bigger than City, they were equal at best(smaller overall in some cases). After the takeover, if that is indeed their measuring stick, how can they even begin to claim City aren't a big club today? It's completely ridiculous and makes no sense by their own logic.