Drug resistance mediated by clonal evolution is arguably the biggest problem in cancer therapy today. However, evolving resistance to one drug may come at a cost of decreased fecundity or increased sensitivity to another drug. These evolutionary trade-offs can be exploited using 'evolutionary steering' to control the tumour population and delay resistance. However, recapitulating cancer evolutionary dynamics experimentally remains challenging. Here, we present an approach for evolutionary steering based on a combination of single-cell barcoding, large populations of 108-109 cells grown without re-plating, longitudinal non-destructive monitoring of cancer clones, and mathematical modelling of tumour evolution. We demonstrate evolutionary steering in a lung cancer model, showing that it shifts the clonal composition of the tumour in our favour, leading to collateral sensitivity and proliferative costs. Genomic profiling revealed some of the mechanisms that drive evolved sensitivity. This approach allows modelling evolutionary steering strategies that can potentially control treatment resistance.