No. this is an IRS imposed limit. to do more than 5 electronically filed returns you either have to cheat (don't know the penalties if caught) or register as an Electronic Return Originator. However, if you are an ERO you can't use Turbotax because that that does not have any of the forms 8879 (differ depending on type of return) that must be submitted with an electronically filed return. That requires one of Intuit's professional versions.
Lacerte or ProSeries.