Turbify stores are highly customizable and, since they come with full-fledged web hosting, many store owners chose to use the hosting portion of their store to host files (images, videos, etc.) or add a blog, or otherwise augment their stores' functionality with custom scripting. These files "live" on a sub-domain of the main store, site.
Leave Hosting on Turbify
You can basically leave everything as is on your hosting account, so everything will still be at site.yourstoredomain.com even after moving the domain to your new Shopify store. To do this, instead of canceling your Turbify account when the time comes, request a "downgrade," so instead of having a full store account you convert it to a hosting-only account.
3rd Party Hosting
Alternatively, you can get third-party hosting from something like GoDaddy or any other hosting provider and set up that new hosting to "live" under site.yourstoredomain.com. You will then need to migrate all your hosting files from Turbify to this other hosting account.
Updates to your Custom Scripts
In either case, if you have custom tools developed for your Turbify store, those tools may need minor or possibly major updates. For instance, if you have tools that make use of the Turbify APIs to communicate with your Turbify store, those will not work once you migrate to Shopify, since the two platforms use vastly different methods to request info from or update info in the stores.