Climate risks increase with mean global temperature, so knowledge about the amount of future global warming should better inform risk assessments for policymakers. Expected near-term warming is encapsulated by the transient climate response (TCR), formally defined as the warming following 70 years of 1% per year increases in atmospheric CO2 concentration, by which point atmospheric CO2 has doubled. Studies based on Earth's historical energy budget have typically estimated lower values of TCR than climate models, suggesting that some models could overestimate future warming. However, energy-budget estimates rely on historical temperature records that are geographically incomplete and blend air temperatures over land and sea ice with water temperatures over open oceans. We show that there is no evidence that climate models overestimate TCR when their output is processed in the same way as the HadCRUT4 observation-based temperature record. Models suggest that air-temperature warming is 24% greater than observed by HadCRUT4 over 1861-2009 because slower-warming regions are preferentially sampled and water warms less than air. Correcting for these biases and accounting for wider uncertainties in radiative forcing based on recent evidence, we infer an observation-based best estimate for TCR of 1.66 °C, with a 5-95% range of 1.0-3.3 °C, consistent with the climate models considered in the IPCC 5th Assessment Report.