The bitterness you are experiencing in your caipirinha almost always comes from two specific factors during preparation: the white central pith of the lime and the technique used while muddling.
First, limes contain a thick, spongy white core running down the center, along with the white layer (the albedo) just beneath the green skin. This white pith is loaded with bitter chemical compounds like limonin. When preparing your lime, it helps significantly to slice off both ends, cut the lime in half lengthwise, and slice out that white central vein before cutting the fruit into smaller wedges.
Second, the physical act of muddling is where most people introduce unwanted bitterness. If you use excessive force, twist the muddler violently against the bottom of the glass, or use a sharp, toothed muddler, you end up pulverizing the peel and crushing the pith. This releases bitter tannins and excessive bitter oils into the juice rather than just the delicate, aromatic citrus oils.
To prevent this, place your lime wedges flesh-side up in the glass, add your sugar over the top, and press down firmly but gently. You want to extract the juice and dissolve the sugar with the aromatic oils from the skin, not shred the peel. A few gentle presses are usually all it takes to get a bright, balanced cocktail without any unpleasant astringency.
