I have noticed this problem with both temple guardians and molten etelia. Sometimes I will win a battle, but when I try to dismiss the battle by clicking the x, I get a message saying that if I quit I will leave the island. Clicking the attack moves has no effect after that (I tried that in case I was wrong about having won). If I refresh the page, I stay on the island but do not seem to get any reward for the battle, even though I cannot fight the guardian/etelia again. 

Edited to add: ... I think what happened was that the refresh hid the results of the battles until I scrolled down. Probably no bug! Also, I feel very silly right now. 

Same! Those happened to me too!

You're not being silly, and it's definitely not a refresh or hiding the results. It's an actual bug because it just happened to me to.

I just battled a Roc boss and after it said I won I tried to exit and it said I would leave without finishing the battle, I clicked it anyway and then I didn't even leave the island, I was still there and I didn't have any reward either.

Thank you for reporting! This is one of the old bugs, it is in our known list here: 

https://beastkeeper.com/thread/1226

Refreshing the page (not accepting the loss) would provide the rewards as expected. The issue was in the logging. 

Thank you for the additional data. It looks like we were finally able to track it and fix! Please let us know if you encounter it again.  

We're extremely sorry for the inconvenience!