So, do these get turned in? And how, if so? I accepted the tasks, but I don't know how to clear them to be able to do more tasks.

are you talking about region 3 tasks? I assume you are
There's only... 7? available tasks, I think 4 for SBO and 3 for Astryl's Tranquility and you can do each one once per day, so you aren't missing much either way
However, if you go to the task log and want to clear a Task to be able to take one of the others without completing it, you click the red X in the bottom right of that task's box.
To complete a task, go to the area that that task dictates, and all the tasks you have for the area will show up in the little progression information box in the lower left of the explore window information.
Once you fulfill the requirements, you return to the task log and the task will have a green tick (I think the tick is in the top right rather than bottom right if I rememeber right, I haven't done one of these in like, two years) that you click to 'turn in'/complete the task

> Find Tinkerbell: SBO, just walk around until you run into her
> Capture Necromancer: SBO, also a 'walk around until you find', once you find him, you fight him
> Collect Apples: SBO, walk around in the orchard/farm area specifically and curb your instinct to flee from everything you run into lest you run away from an apple. Collect the apples when you run into them, and fight lunemara that occasionally attack when you do.
> Defeat... woodsmen I?: SBO, it's a 'defeat x amount of enemies with this word in their name' task
> defeat woodsmen II: AT, same as last one but in Astryl's tranquility
> rare research: defeat water sprites until you get enough water gem 'drops'
> clear flotsam: wander around on lake looking for flotsam patches, basically same as apples but with flotsam and with water sprites attacking you instead.

(You can also pay to unlock more Task slots so you can have more on hand at once, although not really necessary )

Thank you so much! I was clicking everywhere but the green checkmark, I didn't even think that was what I needed to do.