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. Plain Language Summary To keep within 1.5 or 2° of global warming, we need massive reductions of greenhouse gas emissions. At the same time, aerosol emissions will be strongly reduced. We show how cleaning up aerosols, predominantly sulfate, may add an additional half a degree of global warming, with impacts that strengthen those from greenhouse gas warming. The northern hemisphere is found to be more sensitive to aerosol removal than greenhouse gas warming, because of where the aerosols are emitted today. This means that it does not only matter whether or not we reach international climate targets. It also matters how we get there. Key Points Aerosol emission removal can warm the climate by more than 0.5°C Key climate variables are more sensitive to aerosol removal than to GHG increase Regional impacts of 1.5°C warming depend on the balance between aerosol and GHG forcing