Why was my carrot cake very heavy?

If I had to guess, I'd attribute it to the huge amount of stuff that's in the batter, which makes it more like a fruitcake than like a traditional cake.

You only have two cups of flour but you have four cups of carrot, coconut, raisins and nuts. This is definitely going to make a very dense cake and it sounds like it's supposed to.

This is pretty common in carrot cake recipes but it's not required. I recommend that you find a recipe designed to be more cake-y (higher flour to stuff ratio) rather than trying to fix this one. I've been really happy with the carrot cake recipe from Cook's Illustrated/America's Test Kitchen but it's behind a paywall.

The density of a cake or bread largely has to do with the leavening (bubble formation). Cakes like this one are chemically leavened: baking powder or baking soda reacts with acids in the other ingredients to create CO2 bubbles in the batter (like when you combine baking soda and vinegar). There are a lot of factors that determine the end result, including the protein content of the flour and how much the batter is mixed before baking. One significant factor in this case is the physical weight of the added ingredients (carrots, coconut, raisins, nuts). In a plain cake, the pressure from the expanding bubbles pushes the cake upwards, making it rise. If you add a bunch of stuff to the batter, the bubbles have to physically lift those chunks in order to make the cake rise. The bubbles exert the same amount of pressure, but there's more weight, so it comes out with a dense texture.

Another way of thinking about it is in terms of what would happen if you replaced some of the batter with chunks of things. The batter has the active chemicals throughout, and will form bubbles and expand. The chunks contain no bubbles, and just sit there. So with the same amount of cake, you get less bubbles, and hence less lift for the same amount of batter, leading to a dense texture.

Brown sugar will give you a richer flavor than white sugar, but its moisture content is different. Brown sugar's higher moisture content will contribute to a denser, richer cake.

In addition, most butter-based cakes start by creaming the butter and sugar. This not only assists in dissolving the sugar to create a more homogeneous mixture, it incorporates tiny air bubbles. Those air bubbles are the foundation for that light, cakey crumb you're accustomed to in most cakes.

Try replacing some of the oil with butter, and replacing the brown sugar with white sugar (or a combination of the two). Be sure to cream butter and sugar together to get a mixture that looks like light, fluffy frosting. That will give you a head start on a lighter cake.

Actually, the cake turned out very good after 2 days! Became lighter and delicious - my family loves it and asked that I make it again.

I am also going to try making some adjustments as other answers suggested, such as creaming the butter and sugar first. Thinking of eliminating the oil completely. I will let you know the results.