Documentation for JIRA 5.2. Documentation for other versions of JIRA is available too. 
Can JIRA send notifications based on a set issue turnaround time being exceeded? Can it automatically escalate issues that have exceeded a preset turnaround time?
No, not out-of-the box — but this is exactly what services are for. In particular, a Jelly script can be written to find and escalate relevant issues, and the Jelly script can be run every day via a Jelly Service.
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