What does the (apparently nonstandard judging by its absence in the Pilot/Controller Glossary) phrase “turnouts approved” mean when given in the context of a takeoff clearance?
When departing from a towered field pilots may request something other than a "standard" departure. As an example, consider the noise abatement procedures for my home field shown below:
When departing from Runway 14 the standard noise-abatement departure for light aircraft requires you to turn south (heading 190) when you pass the departure end of the runway. This is obviously inconvenient if your destination is to the north, so you can request a northbound departure.
If the tower can accommodate your request they will clear you for the non-standard departure, usually with the phrase "left turn-out approved" or "northbound departure approved", and a heading if necessary.