Starter stays engaged on polaris - Yamaha 2005 Grizzly 660 Auto 4x4

If the soleniod engages and no starter motor run.....the solenoid is bad or bad starter motor brushes.....do not rap on the starter ......the permanent magnets are subject too damage ......replace the starter ..........

is battery charged enough to make starter spin strongly.

Is the engine locked up? check that first. Pull the starter and see if you can turn the flywheel, if not the engine is locked, pull the plugs and check for liquid lock, if the engine does turn the starter may be defective have it tested.

Turn over is the process that an engine makes when it is trying to start. If you twist the key and get nothing then it wont turn over. Google shimming a starter and watch the vids.

Sounds like the key cylinder contacts is at fault on the normal position. you may be able to jump this for a test.
Maybe also confirm the starter is retracting after it initially engages the flywheel. Sometimes heavy corrosion in the starter prevents this from returning and would stall the engine as the gears stay engaged.

Check to see that the pinion gear of the starter is hooting out and engaging the flywheel.
And if it is working it is possible that you have missing tooth/teeth on your flywheel and the starter cannot turn over the engine because of that.

Disconnect the battery and then carefully remove the starter. Take the unit to a local auto parts dealer for a bench test. Starters have two separate functions, one spins the shaft to crank the engine, the other is called a bendix, which pushes the starter gear out to engage the flywheel, and then after the engine engages, retracts it (except in the case of starter/generator combo units which recharge your battery and power your vehicle's electricals while the engine is running) Always best to get a bench test rather than depending on apparent symptoms.

wore out not engaging, remove the cover and alighn the gears, make sure starter wire isn't bare youlll get shocked some times you have to cycle the starter a few times and help it alighn

The starter drive on the starter motor is defective. The teeth on the flywheel could be chewed up too. If you just installed a new starter, you may have to add or subtract clearance spacer metal tabs so that the starter drive engages properly with the flywheel.

Ck for loose, corroded, wire connections or a bad starter solenoid.