The accretion of mass onto supermassive black holes and the conversion of baryons from gas into stars (observed as galaxy stellar mass growth - star formation) follow similar general trends: the cosmic star formation history and the black hole accretion history track each other preserving roughly a ratio of 1000:1, which notably corresponds to the mass ratio between galaxy bulges and black holes observed in the local MBH - relation (Ferrarese & Merritt 2000, Gebhardt et al. 2000, Heckman et al. 2004), whose existence is also often invoked as evidence for co-evolution. Both star formation and black hole accretion history peak around z ~ 1-2 and then decline rapidly towards the present day.
The two growth histories also share further similarities: the most massive galaxies form in intense starbursts at high redshift while less massive galaxies have more extended star formation histories that peak later with decreasing mass (Bower et al. 1992, Thomas et al. 2005, Thomas et al. 2010, Nelan et al. 2005). This `anti-hierarchical' nature is mirrored in black hole growth: the most massive black holes likely grow in intense quasar phases which peak in the early universe, while less massive black holes have more extended, less intense (low Eddington-ratio) growth histories that peak at lower redshift (Ueda et al. 2003, Hasinger et al. 2005, Richards et al. 2006).
This does not imply that star formation rate and black hole accretion rate simply track each other within each galaxy; not every galaxy with a high star formation rate is also a quasar, and vice versa. Rather, within individual galaxies there seems to be an interplay, a co-evolution which regulates the whole galaxy-black hole system to conform to the general trend. How this regulation works is at the heart of the question of how co-evolution works.
The other major piece of evidence often cited in favor of co-evolution is the MBH - relation; since the gravitational sphere of influence of the black hole is tiny compared to the host galaxy, fine tuning of the growth of both is required to produce a tight relation, as is observed. However, recent work by Peng (2007, refined by Jahnke & Macciò 2011) argues that a correlation between galaxy and black hole mass need not be the result of a causal relation at all. Rather, the nature of hierarchical assembly in a CDM universe naturally results in a correlation after a sufficiently large number of mergers via the Central Limit Theorem. This does not mean that the MBH - relation really has a non-causal origin, but rather that its interpretation is not straightforward.