Wait, maybe I am confused. Are you saying your Catholic high school is publicly funded? No Catholic high schools I know of in the US get public funding. Nor do I think they have any kind of local board handing out money. If they do Louisville, KY still has many private Catholic institutions, all of which do better than public schools in their same region.
The main problem is that public schools often do not do it properly.
As for the quality of private schools, I went to college and met kids that went to private schools and already had enough credits earned from high school AP courses to technically be a Sophomore. I had a friend that switched to private-based home schooling and she was a full year behind the curriculum and had to double up and catch up within a year.
Then there is the fact that very often private schools get these results for less money per student than public schools. To see a sample of this from a test case simply look at the DC Voucher program that the Obama Administration killed off to appeas teachers' unons.
http://www.cato-at-liberty.org/2009/03/06/vouchers-vs-the-district-with-more-money-than-god/
$26,555 per student in DC public schools.
An average of less than $6,000 per student in the voucher program.
Results showed students were getting equal or better results in the classroom and parents enjoyed the safer school conditions.