You are not "in" the 24% tax bracket. Some (not all) of your income is. The first $12,000 of income is taxed at 0% due to the personal exemption. The next $9700 is taxed at 10%. Then $29,775 at 12%. Then $44,725 at 22%. Only then are you paying 24%. That gives a total rate for the first $84,200 of 14.95%.
Another issue is that you are only starting work in the middle of the year. So while your full year income might be more than $84,200, the actual amount that you will be paid this year may not be. So even if your full year salary is (e.g.) $90,000, you might only make $60,000 this year.
If you expect to make a lot more than $84,200 even in the partial year, then I would expect future paychecks to withhold more, starting with the second one.
You are responsible for the correct amount, but at least wait for the second paycheck before worrying too much. Assuming you filled out your withholding correctly, they should withhold a reasonably correct amount. I.e. close enough that you won't owe a penalty at the end of the year.
Note also that state and payroll taxes would be in addition to these numbers. So your 13% might jump to 25% or 30% next time, even if your marginal rate is only 24%.