clár is a board. People used to write lists on boards, so clar came to be associated with such lists - a food menu is a biachlár, and a list of choices, like a computer menu, is a roghchlár. Before radio and television, people would go to the theatre for entertainment, and the list of entertainers was called "a programme" in English, and a clár in Irish. "On the programme tonight, we have X and Y and Z". This took on the modern meaning of "a show", so that, instead of a "TV programme" being a list of the available entertainment tonight, it now means a single "capsule" of entertainment - clár teilifíse.
"performance" doesn't really fit into that - the whole point of the the clár was that it was a list of the different performances that would be performed - while "a show" can be substituted for "a programme" or "a performance", I'm not sure if "a programme" can be substituted for "a performance".