Orthogonal Frequency Division Multiplexing (OFDM)
QAM does not require any subcarriers at all, it modulates the carrier signal at the carrier frequency.Both the in-phase (I) and the quadrature (Q) components in QAM are modulated at the carrier frequency, the only difference is that the carrier waves used for modulation are \$90^0\$ out of phase i.e the I-component is modulated with \$\cos(2\pi f_ct)\$ whilst the Q-component is modulated with \$\sin(2\pi f_ct)\$, where \$f_c\$ is the carrier frequency.So the use of QAM does not change the number of subcarriers used in a OFDM scheme.And for clarification in a M-QAM scheme, each symbol will represent \$\sqrt{M}\$ bits, a symbol does not necessarily have to represent 2 bits.

In OFDM, we have \$n\$ different subcarriers so instead of transmitting at the carrier frequency \$f_c\$ we will transmit at a set of frequencies \$\{f_i\} \text{ where } i = 1,..,n\$ and where all frequencies in \$\{f_i\}\$ are very close to \$f_c\$.So if for example we use 4 subcarriers and we have your bitstream of \$0101110011110100\$, we would first map the bitstream to parallel symbol streams \$S_{\# 3} S_{\# 2} S_{\# 1} S_{\# 0}\$ and transmit symbol \$S_{\# 0}\$ at \$f_0\$, \$S_{\# 1}\$ at \$f_1\$, ... e.t.c Because we have 4 subcarriers all 4 symbols will be transmitted in parallel.