On the Droid Turbo (and many other new Verizon specific phones) you have to have HD Calling (sometimes called VoLTE) turned ON to have a data connection while on a phone call
You can turn it on right on your phone. Settings > Advanced Calling > Activate Advanced Calling. It does not cost any extra, its free. The only draw back is: if a call starts on 4G and you move to a 3G signal while on the call the call will disconnect. HD Calling relies on a constant 4G signal to have calls and data simultaneously. Hope that helps.