Hi Arnoud! This is fun :)
I used Wildscreen Arkive to build a larger training set for your LionOrTigerOrBear classifier. Here is the code:
urls = {"http://www.arkive.org/lion/panthera-leo/photos.html",
"http://www.arkive.org/tiger/panthera-tigris/photos.html",
"http://www.arkive.org/brown-bear/ursus-arctos/photos.html"};
trainingset = <| StringSplit[#, "/"][[4]] -> Drop[Import[#, "Images"], 1] & /@urls|>

LionOrTigerOrBear = Classify[trainingset]

The good news is that now LionOrTigerOrBear is correctly identifying the image of the tiger running through some water:

What about the Teddy Bear?

Good job!