OK, we've got 17 entries, so we'll start with a wild-card matchup.
Obviously in evaluating 'successful', quality is important, but so is quantity, all other things being equal. Just to be clear, you're supposed to vote for the actor whom, in your opinion, best personifies having both an extremely successful career as a child actor, and an extremely successful career as an adult actor. How you sort that all out in each case, is up to you.
However, I will provide an estimate of the number of films/tv shows (not including guest appearances) that the person did as a child and as an adult. I'll also list 1 to 3 examples of each, though what I think are good examples might not agree with others. Feel free to go to imdb.com yourself to find out more, or to make your case when you vote, and mention what you think are the appropriate highlights or lowlights of someone's career.
So, without further ado:
Jennifer Connelly vs. Reese Witherspoon
Jennifer Connelly: Child: 7 (Once Upon a Time in America) Adult: 18 (Dark City, A Beautiful Mind, Requiem for a Dream)
Reese Witherspoon: Child: 7 (The Man in the Moon) Adult: 16 (Pleasantville, Election, Legally Blonde)