Why is the foreign tax credit Turbotax gives me significantly less than the foreign tax I reported?

If you go to forms and read through the steps for calculating your credit, you'll see what the form does and why the credit is less.  Basically, you don't get the full tax because of some limitation.

if you are single you can claim up to $300 (married filing jointly $600) without filing form 1116 on passive income - dividends, interest, and royalties. this might avoid the limitation.   if the form is required 

the limitation is if your FTC rate is higher than your effective federal tax rate all you'll get is about the  FTC based on the foreign income taxed at the effective federal rate.