In the case of a TV show, the answer is logical -- editing. When it comes to a live show there's that wonderful insanity behind the big curtain, of naked and half-naked boys & girls running from one spot to the next while some poor soul and throwing clothing on them; then there's the small dressing booths that the VIP types will often have.
Probably the most common practice when it comes to the main performer, is layering outfits that are deliberately break-away. This allows the stage hands (Dressers) to literally pull the outfit off, fluffy what needs to be fluffed and maybe hand the performer a hat or coat. . . with males more than women, the costume change is rarely more than a new sport coat or robe like covering and some kind of head gear.
Costumes worn on stage go through hell and rarely last more than a season or two before having to be replaced due to fabric ware and fasteners no longer holding. Even when cleaned and packed properly, most of that stuff gets smelly and stays that way, so you're more than glad to make the costume changes, and in some cases, must! I know I used to perspire so heavily when on stage, that I had to change shirts and jackets about every 12-20 minutes.
Costume pieces can cost as much as a big illusion in some cases, because of all the special fasteners and reworking that goes into them.