The linewidth–size (σ–R) relationship of the interstellar medium (ISM) has been extensively measured and analysed, in both the local ISM and in nearby normal galaxies. Generally, a power law describes the relationship well with an index ranging from 0.2 to 0.6, and is now referred to as one of the ‘Larson's Relationships’. The nature of turbulence and star formation is considered to be intimately related to these relationships, so evaluating the σ–R correlations in various environments is important for developing a comprehensive understanding of the ISM. We measure the linewidth–size relationship in the Central Molecular Zone (CMZ) of the Galactic Centre using spectral line observations of the high-density tracers N2H+, HCN, H13CN and HCO+. We construct dendrograms, which map the hierarchical nature of the position–position–velocity (PPV) data, and compute the linewidths and sizes of the dendrogram-defined structures. The dispersions range from ∼2 to 30 km s−1 in structures spanning sizes 2–40 pc. By performing Bayesian inference, we show that a power law with exponent 0.3–1.1 can reasonably describe the σ–R trend. We demonstrate that the derived σ–R relationship is independent of the locations in the PPV data set where σ and R are measured. The uniformity in the σ–R relationship indicates that turbulence in the CMZ is driven on the large scales beyond ≳30 pc. We compare the CMZ σ–R relationship to that measured in the Galactic molecular cloud Perseus. The exponents between the two systems are similar, suggestive of a connection between the turbulent properties within a cloud to its ambient medium. Yet, the velocity dispersion in the CMZ is systematically higher, resulting in a scaling coefficient that is approximately five times larger. The systematic enhancement of turbulent velocities may be due to the combined effects of increased star formation activity, larger densities and higher pressures relative to the local ISM.