"Tries to turnover but won't"
Do you mean the starter isn't spinning the flywheel (or barely), or do you mean that it is spinning the flywheel but the motor doesn't fire off and run?
Assuming the former situation, it sounds like an electrical connection or component is adequately carrying amperage when cold but electrical resistance increases with temperature so after the motor has run a while it needs time to cool back off. As mention by others it may be the connections at the solenoid or the negative battery cable to the block which are all hot after the motor has been running. The starter itself can fail in this way, the windings take along time to cool back off as it is mounted right next to the warm engine block and insulated by the engine cover. I bet the motor would restart sooner if you removed the engine cover. Clean up the connections at the block and if this doesn't fix the issue, it is the starter itself.