Most CO2 dissolved in water remains as such, very little if any exists as H2CO3. What little reaction there is, is best written:
CO2(aq) + H2O <=> H+(aq) + [HCO3]-(aq), with the equilibrium far to the left.
making the solution weakly acidic but with few or no H2CO3 molecules formed at all.
The amount of [CO3]2- formed is small indeed as for [HCO3]-(aq) <=> H+(aq) + [CO3]2-(aq) the equilibrium is even further to the left, as expected in the second ionisation of any acid - you are removing a positive charge from a species which already has a negative charge.
When CaCO3 dissolves in carbonated water, the reactions are:
CO2(aq) + H2O <=> H+(aq) + [HCO3]- then
[CO3]2- + H+(aq) <=> [HCO3]-(aq)
Giving as the net reaction [CO3]2- + H2O + CO2 <=> 2[HCO3]-