In this paper, first, the sensitivity of traditional Full-Duplex (FD) communication systems to time variation in the self-interference (SI) channel is demonstrated via performance analysis. It is seen that conventional schemes are not capable of providing efficient operation regarding practical concerns such as spectral efficiency, SI channel aging and learning accuracy, and the contamination of signal-of-interest (SoI). Then, in regard to the aforementioned concerns, a practical FD operation together with a novel iterative SoI-contamination-aware digital SI cancellation (DSIC) algorithm that relieves the sensitivity against time variation considerably is proposed. The proposed scheme, namely <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Turbo</i> DSIC, is practical in terms of training and computational complexity thanks to its two-stage structure such that complex nonlinear parameter learning (NPL) is conducted very rarely in the first stage, and linear processing is performed adaptively at symbol-rate in the second stage, where SoI is iteratively estimated at each data block and eliminated from the received signal to improve SI channel estimation accuracy. Finally, the performance of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Turbo</i> DSIC, under hardware impairments and time-varying (TV) propagation, is evaluated via both computer simulation environment and actual experimental hardware setup in terms of bit error rate (BER) and achievable information rate (AIR) metrics.