Learning symbolic expressions directly from experiment data is a vital step in AI-driven scientific discovery. Nevertheless, state-of-the-art approaches are limited to learning simple expressions. Regressing expressions involving many independent variables still remain out of reach. Motivated by the control variable experiments widely utilized in science, we propose Control Variable Genetic Programming (CVGP) for symbolic regression over many independent variables. CVGP expedites symbolic expression discovery via customized experiment design, rather than learning from a fixed dataset collected a priori. CVGP starts by fitting simple expressions involving a small set of independent variables using genetic programming, under controlled experiments where other variables are held as constants. It then extends expressions learned in previous generations by adding new independent variables, using new control variable experiments in which these variables are allowed to vary. Theoretically, we show CVGP as an incremental building approach can yield an exponential reduction in the search space when learning a class of expressions. Experimentally, CVGP outperforms several baselines in learning symbolic expressions involving multiple independent variables.