Register Telerik HTTP handlers in nested web.config
Register the Telerik AJAX handlers if you cannot modify the root level web.config file.
Add a web.config file in the sub-folder in the web application that you can modify and register the handlers there. Then, use the properties of each control that uses handlers to point them to the new path. For example, for RadScriptManager, this is the HttpHandlerUrl property, for RadEditor you may need DialogHandlerUrl and/or SpellCheckSettings-AjaxUrl, for RadProgressArea you need the AjaxUrl property of RadProgressManager.
The configuration will depend on the handler(s) each control uses and the properties it exposes. You can find the former by monitoring the network traffic through the browser dev toolbar, and you can explore the latter via the intellisense in Visual Studio.
You can find here a basic example of registering the handlers in a folder different than the root.