Three-dimensional strontium ruthenate (SrRuO3) is an itinerant ferromagnet that features Weyl points acting as sources of emergent magnetic fields, anomalous Hall conductivity, and unconventional spin dynamics. Integrating SrRuO3 in oxide heterostructures is potentially a novel route to engineer emergent electrodynamics, but its electronic band topology in the two-dimensional limit remains unknown. Here we show that ultrathin SrRuO3 exhibits spin-polarized topologically nontrivial bands at the Fermi energy. Their band anticrossings show an enhanced Berry curvature and act as competing sources of emergent magnetic fields. We control their balance by designing heterostructures with symmetric (SrTiO3/SrRuO3/SrTiO3 and SrIrO3/SrRuO3/SrIrO3) and asymmetric interfaces (SrTiO3/SrRuO3/SrIrO3). Symmetric structures exhibit an interface-tunable single-channel anomalous Hall effect, while ultrathin SrRuO3 embedded in asymmetric structures shows humplike features consistent with multiple Hall contributions. The band topology of two-dimensional SrRuO3 proposed here naturally accounts for these observations and harmonizes a large body of experimental results.