Basaltic watersheds such as those found in Iceland are thought to be important sites of CO₂ sequestration via silicate weathering. However, determining the magnitude of CO₂ uptake depends on accurately interpreting river chemistry. Here, we compile geochemical data from Iceland and use them to constrain weathering processes. Specifically, we use a newly developed inverse model to quantify solute supply from rain and hydrothermal fluids as well as allow for different mineral phases within basalts to react at different rates, solutes to be removed via clay formation, and some Ca to be sourced from carbonate dissolution. While some of these processes have been considered previously, they have not been considered together allowing us to newly determine their relative contributions.We find that weathering in Iceland is incongruent in two ways. Firstly, solute release from primary silicates is characterized by a higher proportion of Na than would be expected from bulk basalts, which may reflect preferential weathering or some contribution from rhyolites. This Na enrichment is further enhanced by preferential Mg and K uptake by clays. No samples in our dataset (n=537) require carbonate dissolution even if isotopic data (δ26Mg, δ30Si, δ44Ca, and/or 87Sr/86Sr) are included. While some carbonate weathering is allowable, silicate weathering likely dominates. The complexity we observe in Iceland underscores the need for inverse models to account for a wide range of processes and end-members. Given that riverine fluxes from Iceland are more Na-rich than expected for congruent basalt weathering, the characteristic timescale of CO₂ drawdown is likely affected.