Hi Lea,
I started my bub with solids at 4 months (as did a lot of friends of mine - one of which is a GP).
Weaning at 4 months I think is perfectly safe - the only problem is how she is capable of digesting the food. Make sure you research what is safe to give for example cow's milk shouldn't be added to anything and should only be introduced at 12 months as they are able to break down certain proteins.
Carrot sounds like a great start If she has had reflux, be careful not too give her anything too acidic (not that you should this early anyway) and a couple of spoonfuls at a time (as you have been).
Take care