Superconducting fluxonium qubits provide a promising alternative to transmons on the path toward large-scale superconductor-based quantum computing due to their better coherence and larger anharmonicity. A major challenge for multi-qubit fluxonium devices is the experimental demonstration of a scalable crosstalk-free multi-qubit architecture with high fidelity single-qubit and two-qubit gates, single-shot readout and state initialization. Here, we present a two-qubit fluxonium-based quantum processor with a tunable coupler element following our theoretical proposal [DOI: 10.1063/5.0064800]. We experimentally demonstrate fSim-type and controlled-Z gates with $99.55\%$ and $99.23\%$ fidelities, respectively. The residual ZZ interaction is suppressed down to the few kHz level. Using a galvanically coupled flux control line, we implement high fidelity single-qubit gates and ground state initialization with a single arbitrary waveform generator channel per qubit.