The round marker is incremented at the start of the attacker's round only.
See 5.2.1, 4th paragraph - "The Battle Round token will increase by 1 in each subsequent round, at the start of the attacking player's round".
So, each side could potentially get 4 rounds to reinforce.
The battle ends at what would be the start of round 5. See "Draws" under 5.2.3.
I think this should be clarified to identify that the Battle Round token increases only at the beginning of the attacking player's round. As it reads now, the implication is that the token rotates on each round, meaning at the beginning of the attacker's and their opponent's round.
Really all that is needed is to change it to state:
"The Battle Round token will increase by 1 at the start of each subsequent attacking player's round."