Why did I not receive the state tax refund on my bank account but received only the federal tax refund

TurboTax receives no information about refunds. Your state taxing authority probably has a site where you can get information about your refund. 

There is no connection between when you receive your federal refund and when you receive your state refund. They are processed by completely separate agencies, and each one works at its own pace.

To check on your state refund go to the following link and then click on the name of your state.

How do I track my state refund?