I'm an experienced Prince player looking to move up to Monarch, so I decided to pop open your game and take a look. I feel a little qualified to say something.
The one thing I can advise without even seeing the game is this: war is necessary. The AI gets better commerce, cheaper maintenance, more units, free units at the start, etc. but they still can't fight as well as a human. Use this to your advantage. In essence, you have to play like Prussia. Stop thinking of yourself as a country that happens to have a army. Start thinking of yourself as an army with a country.
And now, from my cursory glances around your empire...

I don't know why you are building hammans in your size 2 and 3 cities. You should only build health/happiness buildings when you are within a point or two of reaching the limit. Your cities cities in the north and west can grow to ~7 before they even need to think about building a hamman. Consider building/whipping a granary in both, and then get other infrastructure or soldiers going.

Also, why are those cities in the north guarded by upgraded horse archers when those could be better deployed southwards? Always guard your cities with archers/melee units, and keep mounted units available for striking enemy cities or fighting off the enemy as they cross into your lands. Mounted units do not typically receive defense bonuses (unless UUs), so they do not make good city defenders.

In Istanbul, why do you have a Citizen? Get that guy working in the fields, or working as a specialist. Don't leave citizens if you can avoid it--they do practically nothing, as both priests and engineers more than outweigh that puny +1 hammer, and working in the fields might produce more food (read: more whipping).

As a quick note, the enemy comes in force on Prince. If Cyrus backstabs you, you will lose Bursa quickly. Reinforce that city, and try to keep a border guard of at least 3 units, preferably an upgraded longbow or two, maybe a mace and a mounted unit. Come to think of it, Bursa could use some terrain improvements, like farms. Many of your cities (besides the capital) are lacking in terrain improvements--build more workers to get stuff done). Farms are great because they increase your food, which correspondingly increases your production when you whip your citizens. Since you don't have a lot of high-production mines, that is your best bet for getting stuff done.

I don't mean to sound rude, but your army is in dismal shape. Those nonpromoted warriors are pretty much useless--I would dismiss them, and produce macemen. I didn't see a single mace in your territory. Or longbow. Or crossbow. Those are solid middle-age units that you need for fighting a war--you won't be able to take anything with your current army, especially with cities like Atlanta having a 75% culture defense boost. You need catapults for that, as well as a set of middle age units instead of your current obsolete army if you expect to survive a war.

On the side, you have a nice treasury--I never have that much by this point in the game unless I'm extracting it from an AI. You might be tempted to upgrade some of those warriors to maces, but I would advise against upgrading units without promotions--I just don't think they are worth it.

Now, technology wise, you aren't in great shape either. You are falling behind quickly, but somehow are the only one besides Washington to have maces. You need Construction and Code of Laws--those will allow you to build Catapults and Courthouses, staples for an expanding empire. The Catapults are necessary for breaking down city defenses...build 3 cats, and escort them in with several maces, some city raiders, rest combat + shock/cover/anything else, send a medic, and maybe a handful of spears, longbows, and crossbows. Courthouses are needed to boost your income by cutting costs, and to keep your science rate high while conquering (you don't want to fall to 60% or below for too long, otherwise you will be shafted by your more advanced AI neighbors).

I then looked at your civics--you have Vassalage and Bureaucracy available. Switch to one of them. Vassalage will work well if you get barracks up and start mass-producing soldiers (preferably by beating your populace into submission), while Bureaucracy will greatly boost your capital (which is your largest city in terms of commerce). Either is a solid pick.

I noticed you don't know too much about the map...try to sign an open borders agreement or two (with Cyrus?), and send an old unit around (one of those warriors will do just fine) to map out the world. It helps to know the size and shape of your opponents, where their resources are, etc. That, however, is a detail at the moment. Focus on upgrading your military and getting Construction and Code of Laws.

From the state of your country, it looks like you aren't using the whip very often. For the most part, I will whip out granaries and barracks in my cities so I can start producing experienced troops in several cities quickly (again, usually by whipping--this works great when combined with Vassalage or Theocracy). Also, since your capital is so unhealthy, consider whipping away that problem by rushing some troops. Don't be afraid to sacrifice your population for buildings and troops. I was originally, and the game got a lot easier when I learned to slaughter thousands of citizens to get what I wanted done. I constantly whip in my cities, from the moment I first get Bronze Working until everyone else starts using Emancipation and I can't take the unhappiness anymore.
My course of action would be to whip barracks and granaries in all my cities, if they weren't in place already, and then begin massing longbows and maces. I would switch my research to Construction, and immediately whip catapults once it's completed. Then, mass and attack someone. My choice would actually be Persia due to their weaker standing in power (but you are by far the weakest--pay attention to the Power graph!). However, if you want to keep him as a potential ally, Washington is your next best bet. Be prepared to defend yourself, though, he appears to have a much more massive military.
I don't mean to sound too critical (I'm a critical person, I can't help it), but I just want to offer some pointers that I mostly learned the hard way. I still think you have a chance to win, given your situation, but you have to act quickly to establish a permanent dominance. The key will be beating down an AI with a combo of catapults and maces, which you are able to do at this point because although they are ahead of you, they aren't that far ahead. You might even get a tech out of a peace settlement.
Whew...I originally thought that will be a little shorter of a message, but it turns out I just couldn't stop typing. Just think about it, and see if you can kick it up a little more. Just remember, you can do this, and if this game doesn't work, post another from an earlier year (maybe ~500 AD) and we'll work on that. It definitely took me a few games to get used to Prince, so I know what it feels like.