There are probably records but there are also a few things you have to keep in mind: It's at a state level. You'd have to find out where they did it, likely the city, contact that clerk of court and go through the process of applying to get the info. Even then they likely wouldn't tell you all the details. It's not as simple (to my knowledge) as just going on a website and finding the right database and paying a fee. It's not hard, but also not simple. I think each state handles it differently as well.
One of the requirements in most jurisdictions for a name change is that you make a public statement about that name change. In the past, the general means of that was making a public notice in the nearest major newspaper.
More importantly, there are more facets of your identity that stick with you when you change your name. Bank accounts, etc. To truly divorce yourself from your past identity you have to cut off everything from your past. This is how witness protection works -- the government gives you a new identity from the ground up, and if you have family that comes with you they have to completely separate themselves to your new world as well.
Isabella's parents are very invested in their current identities and are obviously not willing to give them up, so for Bella to successfully change her identity in the US she will have to cut most ties with them. This means limiting financial transactions with them (preferably using cash), establishing entirely new financial instruments, never visiting their house in her own car lest it be photographed, etc. Pulling this off requires a lot of discipline that I'm not sure she has.
The other alternative is to go off-grid entirely by either living in some mountain cabin somewhere, or moving to a smaller country under a new name where the populace does not interact with the English-speaking internet much.
In all of these cases it requires a complete divorce from all social environments where you might be recognized, including the corners of the internet you used to inhabit. All it takes is one single leak for the whole house of cards to come down and for them to get doxxed all over again.
EDIT: People regularly track down people living under new identities when there's enough motivation. Often it's investigative reporters willing to put in a lot of time. The collective effort of a large group of people also can serve this purpose. Basically if enough people are determined to find you, they will, unless you have extremely good opsec. For witness protection folk, they have handlers who help enforce this opsec. Trying to do it on your own is possible, but extremely difficult. If her dad tries to do this for her, he will have to hire handlers to make sure she sticks to the rules, otherwise she will screw up.