How do you define a big club? A Supreme Court judge once said "I can't define pornography but I know it when I see it".
Attendance? Yes but then that can be skewed. Palace have size constraints West Ham don't. Sunderland are a one town club where London has at least 8. So Sunderland are the only game in town.
History? Nothing wrong in being proud of you history but there are many northern towns whose clubs were big before the war but are now perennial lower league participants.
Success? I suspect this is the best measure. A mate once suggested that a big club should win one trophy at least every 10 years. I think this is the best measure.
We are currently in a purple patch but we are not bigger than West Ham we are just better off then them this season. We need another 10 years of better performances than them before we can say we are the 4th club in London.
Go back the the Curbishley era Charlton were bigger than Palace or certainly on a par. I think even the most one eyed Charlton fan would agree we have now pulled ahead of them.