Flexible and iterative baseband receivers with advanced channel codes like turbo codes are widely adopted nowadays, ensuring promising error rate performances. Extension of this principle with an additional iterative feedback loop to the demapping function has proven to provide substantial error performance gain at the cost of increased complexity. However, this complexity overhead constitutes commonly an obstacle for its consideration in real implementations. This paper illustrates the opposite of what is commonly assumed and proposes a complexity adaptive iterative receiver performing iterative demapping with turbo decoding (TBICM-ID-SSD). Targeting identical error rate, the paper shows that for certain system configurations TBICM-ID-SSD presents lower complexity than TBICM-SSD (without iterative demapping). This original result is obtained when considering the equivalent number of iterations through detailed analysis of the corresponding computational and memory access complexity. The analysis is conducted for different parameters in terms of modulation orders and code rates and independently from the architecture for a fair comparison. Considering the proposed adaptive receiver which is able to perform both TBICM-ID-SSD and TBICM-SSD modes, results demonstrate a reduced complexity with TBICM-SSD for high modulation orders. However, for low modulation orders as for QPSK, results show a reduction in arithmetic operations and read access memory up to 45.9% and 47% respectively for using the TBICM-ID-SSD mode rather than TBICM-SSD performing 6 turbo decoding iterations over Rayleigh fading channel with erasures.