Usually not. Your W-4 sizes withholding as if the paycheck were your only income, so the tax on your side income goes unpaid.
Your employer withholds based on the W-4 you filed, and that form treats the paycheck it is attached to as your only income. Side income arrives with nothing taken out. Your withholding was never sized to cover it, so the unpaid tax shows up as a bill in April.
If the side income is self-employment, meaning freelancing, 1099 work, or selling things you make, you also owe self-employment tax of 15.3% on the profit. That applies on top of income tax, not instead of it.
Your withholding may cover it anyway. The test is the safe harbor rule. If your withholding alone reaches 100% of last year's total tax, or 110% if your prior-year adjusted gross income was over $150,000, the IRS cannot penalize you no matter how much the side income adds. You would still owe the difference in April, but with no penalty. The same is true if you will owe less than $1,000 total. The safe harbor calculator runs this test from one pay stub and last year's total tax.
If withholding will not get you there, you have two fixes. You can raise your paycheck withholding by putting an extra amount on line 4(c) of a new W-4. That is the simpler route, because withholding counts as paid evenly through the year even if you raise it in November. Or you can make quarterly estimated payments on the side income directly.
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Sources: IRS Publication 505· Form W-4