We introduce an inductive n-qubit pure-state estimation method based on projective measurements on mn + 1 separable bases or m entangled bases plus the computational basis, with m >= 2. The method exhibits a favorable scaling in the number of qubits compared to other estimation schemes. The use of separable bases makes our estimation method particularly well suited for applications in noisy intermediate-scale quantum computers, where entangling gates are much less accurate than local gates. Our method is also capable of estimating the purity of mixed states generated by the action of white noise on pure states. Monte Carlo simulations show that the method achieves a high estimation fidelity. Besides, the fidelity can be improved by increasing m above 2. We experimentally demonstrate the method on the IBM's quantum processors by estimating up to 10-qubit separable and entangled states. In particular, a 4-qubit GHZ is estimated with experimental fidelity of 0.875.