What are carbonates and bicarbonates and why aren't they studied in organic chemistry?
Strictly speaking, “organic” chemistry is about hydrocarbons – combinations of carbon and hydrogen, and all the rest. But most carbon-only compounds are studied as inorganic.
Don’t worry, if you like it, there’s plenty of opportunity to study them! Generally, carbonates are found naturally as many forms of minerals rather than plant or animal (organic) forms.
There is more than enough (too much?) to study in organic chemistry that the focus on carbonates in inorganic chemistry is not really a problem. Where they are a part of an organic chemical structure, of course the properties of carbonates will be investigated.
Some references to studies of useful organic carbonates are here: https://www.researchgate.net/publication/11512148_Organic_Carbonates and here: http://www.sciencedirect.com/science/article/pii/S0040402014016378