From personal experience, Bard's quality seems to be between GPT-3.5 and GPT-4, closer to GPT-4 if you have to bet. Except when fresh or live data matters, where Bard is clearly superior. (Bard's training data is up to Feb 2023 compared to ChatGPT's Sep 2021, and Bard also gets live data from Google search.)

MMLU benchmark score agrees with this estimate: GPT-3.5 (70.0%), PaLM 2 (81.2%), GPT-4 (86.4%).