Birds are dinosaurs because they are in the clade dinosauria. Their closest living quadrupedal relatives by genetic distance are indeed crocodiles.
But you're right, it is the fossil record that provides the strongest evidence.
But you're right, it is the fossil record that provides the strongest evidence.
Saying that birds have the same DNA as birds is not evidence of much of anything. :)
Eg here's a nice paper discussing Bayesian phylo combined with rexord sampling. https://www.cambridge.org/core/journals/paleobiology/article...
I'm in an adjacent field so take it with a grain of salt.