How does AES bit security increase by round?
A few data points for AES-128. In 2018 Bar-On, Dunkelman, Keller, Ronan and Shamir described a 32-bit attack on 5 round AES (attack complexity here is the maximum of data, memory and computation requirements) and claimed a 99-bit attack on 7 round AES. The 2011 biclique cryptanalysis paper of Bogdanov, Khovratovich and Rechberger claims a 125.4-bit attack on 8 rounds and a 126.18-bit attack on 10 rounds. I'll claim a 128-bit attack against 11 rounds :-)
Based on our current knowledge then, perhaps it looks like an S-curve? Of course cryptanalytic effort has focussed strongly on larger numbers of rounds and I'd be hesitant to hypothesise any security vs. rounds function based on this.
ETA: The quoted figures for 8 and 10 rounds were for a related key biclique attack. The current best generic attack for 10 rounds is $2^{126.01}$