Let's get the answer out of the way in one line, because you came here for it.
No. Capital gains are not taxed inside an IRA.
You can buy a stock at $20, sell it at $200, and owe nothing. No form arrives. Nothing goes on your return. The IRS does not care, does not ask, and does not want to hear about it.
That sounds too good, and here is where it gets interesting: it is too good, in a very specific way that almost nobody explains.
Because the IRA does not eliminate the tax. It reshapes it. Every dollar of gain inside a traditional IRA gets converted, quietly and permanently, from capital gain into ordinary income.
And ordinary income is taxed at higher rates than capital gains.
So the correct answer is not "IRAs are tax-free." The correct answer is "IRAs trade a series of small taxes now for one differently-shaped tax later," and whether that is a good deal depends on things worth understanding.
Let's do it properly.