Quantum speed limits (QSLs) impose fundamental constraints on the evolution speed of quantum systems. For systems with time-independent Hamiltonians, the QSLs are intricately linked to the energy spectra of the initial state, which signify the generalization of the time-energy uncertainty relation. Based on arbitrary-order norms of the Hamiltonian, dual energies from inverted spectra, and the standard deviation of energy, different types of QSLs have been proposed, revealing three dynamical parameter regimes of quantum state evolution. Despite the theoretical advancements, a comprehensive experimental examination remains elusive. In this research, by introducing unified bounds suitable for all evolution times, we present the experimental verification to the QSLs in various scenarios with a superconducting circuit. We explore the dynamical evolution with regard to each parameter regime as well as the critical point by carefully engineering the initial-state population of the superconducting resonator. We test our unified bounds over finite-level states and infinite-level classical and nonclassical Gaussian states, which validates the criteria for the QSLs. Our work substantially deepens our comprehension of unitary quantum evolution.