In this paper, we explore a fractal model of the universe proposed by
Calcagni [JHEP{\bf03}(2010)120] for a power-counting renormalizable field
theory living in a fractal spacetime. Considering a timelike fractal profile,
we derived field equations in fractal cosmology, in order to explore the
structure formation and the expansion history in fractal universe. Numerical
investigations based on matter power spectra diagrams report higher structure
growth in fractal cosmology, being in contrast to local galaxy surveys.
Additionally, according to the evolution of Hubble parameter diagrams, it can
be understood that Hubble constant would decrease in fractal cosmology, which
is also incompatible with low redshift estimations of $H_0$. So, concerning
primary numerical studies, it seems that fractal cosmology is not capable to
alleviate the tensions between local and global observational probes. Then, in
pursuance of more accurate results, we constrain the fractal cosmology by
observational data, including Planck cosmic microwave background (CMB), weak
lensing, supernovae, baryon acoustic oscillations (BAO), and redshift-space
distortions (RSD) data. The derived constraints on fractal dimension $\beta$
indicate that there is no considerable deviation from standard model of
cosmology.
