In the US, the standard course sequence (e.g. at a good state university) is two years of calculus, diffeqs, and linear algebra (all taught as on-paper computation) concurrently with a course in discrete mathematics. The discrete mathematics course often doubles as an introduction to proof (as is apparently the case at UCI). Year 3 typically covers proof-based analysis, algebra, and linear algebra and some electives. Year 4 is typically electives.
At a fancy school, you can often take proof-based honors versions of Year 1-2 courses but you still may not get to skip over all of Year 3. Think: calculus using Spivak and real analysis using Rudin.
At Harvard, you can take Math 55, which is essentially Year 3 above (plus complex analysis) in Year 1.