Is it correct to say I'll turn on my TV when sports are on?

Yes. Originally, the grammatical rule was to never end a sentence with a preposition. However, this has become obsolete in modern English usage.

Another example, " Go on ! " meaning continue with your ......

Yes. There is no rule against ending an English sentence with a preposition. That is a genteel error foisted on schoolchildren by educationists. It is Latin sentences that may not end with a preposition. Jamming English grammar into a Latin straitjacket is something no idiomatic speaker of English will put up with.