(If it is clear that you are not talking about a person, but about a piece of electrical or electronic equipment, "turn on" is perfectly normal, and doesn't carry any hidden ideas. Teenagers do not giggle if you say "turn on the lights.")
In the United States, and referring to electricity, "turn on" and "switch on" are synonyms. I can't think of any difference in meaning at all.
"Turn on" emphases the physical action, "switch on" emphasizes the device itself (the switch).
"Switch" emphasizes the idea of something that can be completely off or completely on, but nothing in between.
If we are talking about a dark room, we say that we "turn on the lights."
In the special case of a computer, it is common to say "turn on the computer" or "switch on the computer," but it is also common to say "power up the computer" or "fire up the computer."
Water faucets are controlled by a valve and a handle which turns. The valve is not called a "switch." You can "turn" the water on or off, but you cannot "switch" it on or off.
My guess is that the usage "turn on the lights" might go all the way back to the age of gaslights, in which you turned a handle to open or close a gas valve.