  We establish uniform error bounds of time-splitting Fourier pseudospectral
(TSFP) methods for the nonlinear Klein--Gordon equation (NKGE) with weak
power-type nonlinearity and $O(1)$ initial data, while the nonlinearity
strength is characterized by $\varepsilon^{p}$ with a constant $p \in
\mathbb{N}^+$ and a dimensionless parameter $\varepsilon \in (0, 1]$, for the
long-time dynamics up to the time at $O(\varepsilon^{-\beta})$ with $0 \leq
\beta \leq p$. In fact, when $0 < \varepsilon \ll 1$, the problem is equivalent
to the long-time dynamics of NKGE with small initial data and $O(1)$
nonlinearity strength, while the amplitude of the initial data (and the
solution) is at $O(\varepsilon)$. By reformulating the NKGE into a relativistic
nonlinear Schr\"{o}dinger equation, we adapt the TSFP method to discretize it
numerically. By using the method of mathematical induction to bound the
numerical solution, we prove uniform error bounds at
$O(h^{m}+\varepsilon^{p-\beta}\tau^2)$ of the TSFP method with $h$ mesh size,
$\tau$ time step and $m\ge2$ depending on the regularity of the solution. The
error bounds are uniformly accurate for the long-time simulation up to the time
at $O(\varepsilon^{-\beta})$ and uniformly valid for $\varepsilon\in(0,1]$.
Especially, the error bounds are uniformly at the second order rate for the
large time step $\tau = O(\varepsilon^{-(p-\beta)/2})$ in the parameter regime
$0\le\beta <p$. Numerical results are reported to confirm our error bounds in
the long-time regime. Finally, the TSFP method and its error bounds are
extended to a highly oscillatory complex NKGE which propagates waves with
wavelength at $O(1)$ in space and $O(\varepsilon^{\beta})$ in time and wave
velocity at $O(\varepsilon^{-\beta})$.
