Your understanding is incorrect.
1. Your long term capital gain (LTCG) is part of your AGI calculation.
2. LTCG are taxed at certain rates, but which rate depends on your other income. LTCG are added to taxable other income .
Your regular income is first taxed at the lower rates (10 & 12%). Your LTCG are added to other income to determine your next higher tax bracket. But rather than being taxed at the higher rate (next usually 22%), they are instead taxed at the lower LTCG rate (starting at 15%).
For details, see the Qualified Dividends and Capital Gains worksheet (abbreviated "Qual Div/Cap Gn" on the forms list)
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