Some TV shows and even movies show prominent disclaimers, but I’ve only seen these for nonfiction (or reality TV) like MythBusters, and maybe strongly objectionable fiction (TV-MA). For an example of a movie with a warning like this there’s Jackass:
WARNING: The stunts in this movie were performed by professionals. So neither you nor your dumb buddies should attempt anything from this movie.
Media intended for children doesn’t tend to include disclaimers like this as a disclaimer sets a certain tone and it could scare off parents. (Craft shows may have a verbal disclaimer to “get adult supervision”, but this is different because the point is to try the activity at home.)
For a rather different example, the rereleases of 70s-80s Sesame Street included a disclaimer that the content is “intended for grownups”. Though originally intended for children, the episodes depicted things like children riding bikes without helmets, a girl going to the house of a man she just met to get ice cream, and Cookie Monster smoking then eating a lit pipe. The purpose of the disclaimer is clear: the studio didn’t feel it was appropriate anymore for a young audience and wanted to make sure that no parents were caught off guard by this.
Other Warnings: Ratings
Most movies and TV don’t have a disclaimer per se but potentially objectionable content should never be a surprise due to everything having a rating (eg G, TV-14, R). The rating appears in much of the promotional content in addition to being shown directly before the content airs. In addition, there is often a short description of why, such as “cartoon violence”. The parent (or adult viewer) is expected to pay attention to this and act accordingly.
A company that sells insurance (including some for films) explains what’s happening behind the scenes of moviemaking. You’re right that someone reviews the movie, looking for any potential legal problems:
An expert review of the final film, by a media attorney and/or experienced media liability insurance professional, can help to prevent potential problems from ever arising.
They give the example of a documentary where an attorney recommended that the filmmakers obtain permission to use footage relating to a murder from the families of the victims to prevent an emotional distress lawsuit.
Additionally, all movies have insurance of some type. To again quote from the insurance company:
It's impossible to distribute a film without insurance, since distributors require it, so insurance is critical from that standpoint.
The important considerations in purchasing insurance, therefore, are covering the most important exposures, the dollar amount of coverage, the insurance company and broker and value (the coverage received for the amount of premium paid).
It looks to me like not all insurance plans would cover “tried it at home” lawsuits, but a company that thought that was a possibility would go for better coverage. Lawsuits are extremely expensive.