We present Q-U-I JOint TEnerife (QUIJOTE) intensity and polarisation maps at 10-20 GHz covering a region along the Galactic plane 24° ≲ l ≲ 45°, |b| ≲ 8°. These maps result from 210 h of data, have a sensitivity in polarisation of ≈40 μK beam-1 and an angular resolution of ≈1°. Our intensity data are crucial to confirm the presence of anomalous microwave emission (AME) towards the two molecular complexes W43 (22σ) and W47 (8σ). We also detect at high significance (6σ) AME associated with W44, the first clear detection of this emission towards a supernova remnant. The new QUIJOTE polarisation data, in combination with Wilkinson Microwave Anisotropy Probe (WMAP), are essential to (i) determine the spectral index of the synchrotron emission in W44, βsync = -0.62 ± 0.03, in good agreement with the value inferred from the intensity spectrum once a free-free component is included in the fit; (ii) trace the change in the polarisation angle associated with Faraday rotation in the direction of W44 with rotation measure -404 ± 49 rad m-2 and (iii) set upper limits on the polarisation of W43 of ΠAME < 0.39 per cent (95 per cent C.L.) from QUIJOTE 17 GHz, and <0.22 per cent from WMAP 41 GHz data, which are the most stringent constraints ever obtained on the polarisation fraction of the AME. For typical physical conditions (grain temperature and magnetic field strengths), and in the case of perfect alignment between the grains and the magnetic field, the models of electric or magnetic dipole emissions predict higher polarisation fractions.