Limiting global warming to 1.5 or 2.0°C requires strong mitigation of anthropogenic greenhouse gas (GHG) emissions. Concurrently, emissions of anthropogenic aerosols will decline, due to coemission with GHG, and measures to improve air quality. However, the combined climate effect of GHG and aerosol emissions over the industrial era is poorly constrained. Here we show the climate impacts from removing present‐day anthropogenic aerosol emissions and compare them to the impacts from moderate GHG‐dominated global warming. Removing aerosols induces a global mean surface heating of 0.5–1.1°C, and precipitation increase of 2.0–4.6%. Extreme weather indices also increase. We find a higher sensitivity of extreme events to aerosol reductions, per degree of surface warming, in particular over the major aerosol emission regions. Under near‐term warming, we find that regional climate change will depend strongly on the balance between aerosol and GHG forcing.