When I get to the part where it says to enter my expenses for my small business, it keeps jumping over the part where I actually get to enter the expenses

Are you using TurboTax Online?  If so, you need to use the Premium edition to enter your business expenses as Deluxe only handles business income.   Also try clearing your cache and deleting cookies.   How to Clear Cache  and also, how to Delete Cookies

Where do I enter my self-employment business expenses, like supplies and  vehicle mileage?

What software are you using---we cannot tell.  If you are using online TurboTax you need Premium in order to prepare a Schedule C for your expenses.

If you have desktop download, any version can prepare a Schedule C.