We study constraints on allowed reionization histories by comparing predictions of a physical semi-numerical model with secondary temperature and polarization anisotropies of the cosmic microwave background (CMB). Our model has four free parameters characterizing the evolution of ionizing efficiency $zeta$ and the minimum mass $M_{mathrm{min}}$ of haloes that can produce ionizing radiation. Comparing the model predictions with the presently available data of the optical depth $tau$ and kinematic Sunyaev-Zeldovich signal, we find that we can already rule out a significant region of the parameter space. We limit the duration of reionization $Delta z=1.30^{+0.19}_{-0.60}$ ($Delta z < 2.9$ at $99%$ C.L.), one of the tightest constraints on the parameter. The constraints mildly favour $M_{mathrm{min}} gtrsim 10^9 mathrm{M}_{odot}$ (at $68%$ C.L.) at $z sim 8$, thus indicating the presence of reionization feedback. Our analysis provides an upper bound on the secondary $B$-mode amplitude $D_{l=200}^{BB}<18$ nK$^2$ at $99%$ C.L. We also study how the constraints can be further tightened with upcoming space and ground-based CMB missions. Our study, which relies solely on CMB data, has implications not only for upcoming CMB surveys for detecting primordial gravitational waves but also redshifted 21 cm studies.
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