Can I still get accepted into a good university?

[Masha]

So I want to get accepted to a really good university. However, I did poorly on my first two years of high school. My grandmother died freshman year she raised me and my mom and I just kept arguing. Then my sophomore year I was doing way better at my school but I got jumped right after first semester and my mom ended up transferring me to a school with a completely different curriculum and with people I didn't get along with at all. Anyway, so I did really really bad freshman and sophomore year, however this year I'm doing way better at my new school.. although it's not the best school around. If I get a good ACT stores and good grades my junior and senior year can I still get accepted to a good university? ;[ Also , I don't participate in high school activities besides student government, but I participate in a lot of extracurricular activities outside of school. Will that help me at all?