In several previous questions I covered the list of theories that have been left behind by new observational data. The single biggest piece of evidence is the expansion of the universe. No astronomers debate this evidence anymore, only the magnitude of this effect which then establishes the age of the universe.
The next piece of evidence is the existence of a smooth cosmic microwave background radiation, sometimes called the 'fireball' radiation. Alternate theories of the universe either ignore this observational fact, or try to cloud its existence as the consequence of some new type of physical phenomenon which has never been observed previously.
The existence of a 'universal' abundance ratio for helium and deuterium relative to hydrogen is also a key piece of observational data that only Big Bang cosmology has a simple answer for.
No one has ever found REALLY OLD STARS that are more than 15 billion years old. There are no stars 20, 25, 30 .... billion years old even though they ought to be readily observable. The simplest explanation is that the universe was 'born' a finite number of years ago as is indicated by its current expansion rate.
Finally, the night sky is black. This also means that the universe is finite in some way, and the best explanation is that stars have only been around for 15 billion years or less, otherwise the universe would be so vast that every line of sight on the sky would end up on the surface of some distant star.