1. Wrong: Elise and Nathalie danced like they could have been in Chicago's Joffrey Ballet Company.
Better: Elise and Nathalie danced as if they could have been in Chicago's Joffrey Ballet Company.
Explanation: The like in the original sentence can't introduce a dependent clause.
2. Wrong: Vladimir wanted to eat like a hundred soft-serve ice creams.
Better: Vladimir wanted to eat approximately a hundred soft-serve ice creams.
Explanation: Like doesn't mean approximately.
3. Wrong: On their camping trip, Liam and Oscar brought supplies like a tent, sleeping bags, and a water filter.
Better: On their camping trip, Liam and Oscar brought supplies, such as a tent, sleeping bags, and a water filter.
Explanation: If they brought supplies like those named, then they didn't bring the actual objects named. Such as includes the named objects.
4. Wrong: Like my mother always says, "Treat other people like you want to be treated."
Better: As my mother always says, "Treat other people as you want to be treated."
Explanation: Like can't begin either dependent clause.