That's why everybody doing experiments in this space should either 1) be using the OpenAI Playground, or 2) using the API, and not using ChatGPT.
The API for them is already structured conversationally - you don't provide a single prompt to complete, you provide a sequence of prompts of different types ("system", "assistant", "user"); however they mix those together on the backend (some ChatML nonsense?), the models are fine-tuned to understand it.
That's what people mean by "API access to ChatGPT". Same models, but you get to specify system prompts, control the conversation flow, and don't have to deal with dog-slow UI or worry about your conversation getting "moderated" and being blocked with a red warning.
(The models themselves are still trained to refuse certain requests and espouse specific political views, but there isn't a supervisor looking at you, ready to step in and report you to the principal.)
Don't need you to explain how the APIs work... and it seems that GPT3.5 UI is doing something else, using the "text-davinci-002-render-sha" model, just look in the browser dev tools. I'm not sure the UI is using anything beyond the smallest context size for GPT4 either, give the output is cut off earlier than 3.5 and it too loses focus after enough messages in a conversation...
GPT3.5-turbo/GPT4 is way ahead in instruct tuning and does not require such verbal gymnastics.