Even if crossbows were the armor-piercing wonders often claimed, they were not exactly common among peasants (like firearms are in the modern era, in many areas). In fact, I can't think of anywhere that suggested anyone but the military would own a crossbow. After all, normal bows are perfectly good for any kind of non-peasant-uprising shooty stuff a peasant would do; awfully suspicious, neh?
Anyways. I started a game of Civ V to refresh my memory and put some of your tips into practice, but you know how it goes; you're just playing a bit and before you know it it's midnight and you've barely started what you came in to do.
I'm playing Nebuchadnezzar on a small Earth-style map. I wound up in North America next to what I assume is supposed to be one or more of the Great Lakes. (I thought all civs started in the Old World?) I spread out a bit, exploring, but was restricted by mountains blocking most routes and ocean the rest. I researched Optics to explore Alaska, Greenland, Iceland, South America, and part of Siberia. During this last venture, I came across some people who knew Napoleon, currently at the bottom of the pack, but he later moved up. (I'd like to imagine that his increase in score was because he could casually mention that he knows that guy at the top of the score list, score as high as any two other players, largely due to Science and Wonders I'd imagine.) Anyways, I encountered four city-states: Florence about where New Orleans would be in the real world, Kathmandu in the heart of the Amazon, Singapore in what we would call Argentina, and Budapest's borders before my Siberian scout got killed by barbarians while running from a barbarian ship. Florence complained about Kathmandu, Kathmandu complained about Singapore, Budapest complained about Dublin, and I decided to invade Kathmandu.
I sent two Bowmen to Kathmandu, and discovered two quirks:
1. I should have realized this, but ranged units can't range through woods or the like. Great, this whole thing is going to take even longer.
2. One of the problems I'd been having with capturing cities? They heal faster than I can do damage to them! Here I am, two legions of bowmen raining flaming arrows from the sky, probably daily, for centuries, while the rest of my army was being produced and crawling down North America and the Gulf of, um...dunno what we'd call it, what's halfway between Florence and Kathmandu, Iran? The Gulf of Iran? to the Amazon Rainforest, and nothing was coming of it. Mind, I had not only two archers, but the Statue of Zeus (+15% City Attack), one of the archers had leveled a couple of times, once taking the city raider bonus (+25%), and I was in a Golden Age (almost certain that gives me something). I'm pretty sure there were other bonuses I'm forgetting, but I can't remember. In short, the cards were stacked in my favor, and I still couldn't cause any damage!
"Attack with more than two Archers, GWG!", you say? Firstly, I was attacking with the Babylonian Bowmen, which in addition to being alliterative, have more strength than a normal archer. Not sure if that applies to making their attacks stronger, but it certainly makes them tougher. Even with that, and a near-death promotion-heal, one archer still managed to get splattered before I could get reinforcements there--a bit of a chore, since I was basically shooting off units to arrive a dozen turns or so in the future, guessing what units would have arrived or died. In older Civilization games, this wouldn't matter, because of unit stacking. I did get backup, though--a Swordsman, some more Bowmen, a Warrior I had one of my little cities build--but they didn't do much except maneuver into position and, in the swordsman's case, get brutally killed after dealing some damage to the city. Which, naturally, got healed within a couple turns.
I got a Great General on the last turn I played (100), which might help. Less helpful: Kamehameha, evidently the second- or third-strongest player, showed up to my doorstep with a Maori warrior embarking (he must have Astronomy), claimed Florence was his sphere of influence despite being next door to me after I allied with it, and declared that he would protect Kathmandu after I was well underway with my "reinforcements" arriving.
In short: Even with as many bonuses as I could find, attacking undefended city-states (cultural city-states, even!) with a couple units not only dooms you to failure, but can't do more damage than the city heals each turn! I wish I was kidding here. Moreover, if the city isn't surrounded by open fields, it's pretty much impossible to get much more than half a dozen units attacking, and apparently damage is rounded down or something because they don't seem to take more than one from attacks, so you'll need about half of that just to overcome the city's healing!
Seriously, why do cities heal so much? And deal so much damage? Stalingrad was impressive, but I don't think it was twice as tough and thrice as good at self-repair as a decent military unit of the time. Especially considering how close they came to losing in a single turn (and how their loss would have been inevitable if Russia hadn't sent in some real units to help).