So I have a Cog, I have a land unit on the same tile as the Cog. The land unit now has the "Embarked" status. It also has the Lost at Sea status (so does the Cog). I'm trying to get this land unit across a couple turns of ocean to an island. The Cog can do it just fine and arrives at the other side a little bit damaged. However as soon as I move the two together out into the ocean I get the warning that I will lose the unit if it is there one more turn.

So I'm not sure what is going on. I lose the land unit that is embarked ON THE BOAT but I don't lose the boat that it is on!??!! Why would it jump off the boat!?!??

I don't get it. How am I supposed to transport units if they jump off the ship they're on and suicide in the ocean?

There are two different kind of naval units: battle ships and transports. When you embark your army it converts to transport ships that come from nowhere (you don't have to build it). Even if you have military ships (like quadriremes and cogs) on same tile, it doesn't affect embarked units. 

It's very unclear since there are no any indication what kind of transport embarked units use, it just a "regiment in a water" - at least, I don't know where it shown. To be able to transport land units over long water ranges, you must research Caravels, don't remember exact tech for it.

I would like to know the answer to this too. I lost 4 units for just 2 turns of deep water. I reloaded the save and went around a longer way across one ocean tile. What is the downside of embarking units just by themselves? Can they only do shallow water? I noticed that they could embark on accident, but I made them go back to land because I was scared to lose them and used the Cog instead.

Drag your mouse over the boat perks and it explains the ship flaws with ocean tiles. They start by saying something like they have one turn they can stay in ocean and then they die. If you pick phoenicians though they have a special boat that can last a few turns in ocean (1 tech tier earlier). In tech tier II all ships can last a few turns in the ocean rather than outright die. So on and so on... future boat tech (transport or not) follow different techs that upgrade the boat quality.

Anyhoo... you get the "lost at sea" perk on the unit until they get back to a coastal tile. If they end turn again on ocean they either take damage or die, pending your boat tier technology.

You need to research Three-Masted Ships to get your military units across deep water without a deep water penalty.

You need to research Naval Artillery to get real boats across deep water without a deep water penalty.

Once you research Three-Masted Ships, any military unit you have will automatically upgrade to a caravel without being in your territory and without you paying for it.

Once you research Naval Artillery, any real boat you have will have to return to your territory to upgrade to a carrack, and you'll have to pay for the upgrade.

Does this seem nonsensical? Yes. It's one of the more serious flaws with naval units in this game.

I think the issue is more one of UI than anything else.  There is no real indication of what is going on.

I want to say with tutorials turned on it pointed it out on the UI the first time it happened to me. It pointed out the "lost at sea" blip and said why it occurs and how to get rid of it.

Sounds like a classic example of army vs navy dispute not playing nice together. Acting like one is more important over the other taking healthy competition to a new level.

So here is another issue of Amplitude not thinking this through.  I built a Settler unit.  Moved them out into the Ocean after having researched Three-masted Ship.  The first turn they were in the Ocean they had the Lost at Sea flag.  The second turn it disappeared.  More misleadin info for the user around this feature.  

This just hasn't been buttoned down yet - I'm sure they'll get to it.  I have bigger issues than this with the game but this is one that needs some info design.

Yikes, I finally saw that myself as well on my first game. I bet they will fix it eventually. Probably needs to be fixed sooner than later. Naval warfare does not seem to have much appeal right now. That said, I have yet to make a game on a mostly water map to find out how hindering this is.