Bandwidth is defined at positive frequencies. So for the lowpass (baseband) signal in your first figure, the bandwidth equals its upper cut-off frequency, whereas in the bandpass case (your second figure), the bandwidth equals the upper cut-off frequency minus the lower cut-off frequency. So, as you've correctly observed, the bandpass signal's bandwidth is twice the bandwidth of the baseband signal.
It's a misunderstanding that you "can't transport negative frequencies". You always do, as long as you send real-valued signals. Real-valued signals always have a (conjugate) symmetric spectrum.
It's true that this type of amplitude modulation (double-sideband AM) with twice the bandwidth of the baseband signal is inefficient. There are two options to avoid this inefficient use of bandwidth. First, use single sideband modulation (as mentioned in Jason R's answer), or, second, use a complex baseband signal, i.e., use two signals and orthogonal carriers. This is called quadrature modulation, and it uses the same bandwidth as conventional (double-sideband) amplitude modulation, but it transmits two signals instead of one.