Abstract The smooth spin-down of young pulsars is perturbed by two non-deterministic phenomenon, glitches and timing noise. Although the timing noise provides insights into nuclear and plasma physics at extreme densities, it acts as a barrier to high-precision pulsar timing experiments. An improved methodology based on Bayesian inference is developed to simultaneously model the stochastic and deterministic parameters for a sample of 85 high-$\dot{E}$ radio pulsars observed for ∼ 10 years with the 64-m Parkes radio telescope. Timing noise is known to be a red process and we develop a parametrization based on the red-noise amplitude (Ared) and spectral index (β). We measure the median Ared to be $-10.4^{+1.8}_{-1.7}$ yr3/2 and β to be $-5.2^{+3.0}_{-3.8}$ and show that the strength of timing noise scales proportionally to $ν ^{1}|\dot{\nu }|^{-0.6± 0.1}$, where ν is the spin frequency of the pulsar and $\dot{\nu }$ its spin-down rate. Finally, we measure significant braking indices for 19 pulsars, proper motions for two pulsars and discuss the presence of periodic modulation in the arrival times of five pulsars.