Thank you for writing.
I would suggest that you use RadTextBoxControl
that supports the desired feature. Moreover, the scrollbars that appear for RadTextBoxControl are our own, so they match our styles, while in the case of RadTextBox, we are hosting the standard TextBox control, hence the functionality and ScrollBars there are the standard ones. Here is a sample code snippet that will allow you to achieve the desired behavior:
I hope this helps.
All the best,
the Telerik team