Oxides of carbon, carbonates and hydrogen carbonates are covalent compounds but not organic.
The distinction between "organic" and "inorganic" compounds has been retained through the present. The modern meaning of "organic compound" is any one of them that contains a significant amount of carbon - even though many of the "organic compounds" known today have no connection whatsoever with any substance found in living organisms. There is no "official" definition of an organic compound. Some text books define an organic compound as one containing one or more C-H bonds; others include C-C bonds in the definition. Others state that if a molecule contains carbon it is organic.
In summary, most carbon-containing compounds are organic, and most compounds with a C-H bond are organic. Not all organic compounds necessarily contain C-H bonds