Starting in algebra 1 in 9th grade and completing calculus BC in high school is not “behind” in math (even for math majors). According to http://www.math.brown.edu/~calcplacement/ , Brown expects most students to place into (single variable) calculus, rather than be far advanced beyond it. https://bulletin.brown.edu/the-college/concentrations/math/ indicates that Brown’s math department expects math majors to take multivariable calculus and linear algebra (i.e. does not expect them to be advanced beyond that).
That said, try not to get fixated on a super-selective dream school. What you really want to do is build a list, starting from colleges where you are assured admission and assured affordability where the math department offers sufficient upper level pure math courses (real and complex analysis, abstract algebra and number theory, geometry and topology, logic and set theory, numerical analysis, etc.) to fulfill your desire to study pure math.
If you are from a self-described “upper middle class” family, be sure to talk to your parents about what they can afford to contribute for your college, and run the net price calculator on each college’s web site to get an idea of how much it will cost after financial aid. Many self-described “upper middle class” parents will have a high expected parent contribution at many colleges, but may not actually be able or willing to pay that much.