Beleriphon
Totally Awesome Pirate Brain
The ancient city of rome is bigger than waterdeep at about 14km (8.6miles) across
It also had running water, public baths, public latrines, & more thanks to the Aquaduct... Waterdeep has wells.. More importantly Rome was the center of an organized empire that managed trade, conducted war, expaded itself, built roads, maintained roads etc. Waterdeep not so much. The Roman Empire had better roads than FR & the city of Rome could not have existed at the timewith out those roads. I may have overlooked it, but nobody in Faerun is building or maintaining roads llike the romans did. Rome had that population because it did all those things. Other huge trade hub cities at the time were things like (Alexandria, Antioch, Carthage, Ephesus, Salona etc.) had populations of about a few hundred thousand & waterdeep should be similar at best. Faerun being isolated nations little more than manorial towns leaning so heavily on the bones of past empires really hurts it when it comes to plausibly supporting cities like waterdeep.
That's all true, but Waterdeep is also has magic, its not quite Eberron level magic, but its not exactly uncommon by and large. Despite the veneer of faux medival setting, Waterdeep is much more like a late renaissance city than medieval.
Its also ignoring the Lord's Alliance and other institutions that more or less act as nation state federations.
Your target choice of the 1900ish population explosion of those cities ignore what happened with high density construction capabilities between 1885 & 1908. The tallest skyscraper between 1785 & 1885 was 52 feet tall. From 1885 to 1889 it was nearly 3 times that at 138 feet tall. Then again 1889-1908 it was 550feet tall... They just kept growing in height from there. Population density exploded because people started building up rather than out & more importantly things like freight trains& soon after cars/trucks became ubiquitous for bringing produce, raw materials, & finished products into, out of, & across those booming cities.... faerun doesn't have any of that.
Again, true but you're ignoring that in 1801 the UK was a largely agrarian society and the Industrial Revolution was just barely getting started. So between 1700 and 1801 the population had nearly doubled to 1 million people.
Having a plausible population in Waterdeep does not hurt Waterdeep or FR, in a lot of ways it actually helps it by removing some of the "don't look over here" handwaiving the OP touched on.
A population of over 1 million people isn't unreasonable for Waterdeep, it just requires understanding 1) magic suffuses the city to some degree and 2) the Golden Fields outside the city are permanently blessed by Chauntea (so they alwasys produce bumper crops, and crops grow faster).