Should you always suggest the next date or take turns?

Start by separating two things: how warm she is in the chat, and how much she puts into logistics. Someone can be funny, responsive, and clearly enjoying the conversation while still never proposing a date. That usually means she is comfortable letting the other person set the pace, not that she has lost interest. Look at whether she asks you questions about your week, remembers details, and brings up things you said earlier. If she does, the warmth is real and the planning gap is about habit or caution.

Now look at what happens when you do propose. If she says yes with enthusiasm and suggests a time or a place within the same reply, she is engaged and simply not initiating. If she says "maybe" or "we'll see" and leaves it open, she may be unsure how much she wants to commit, or she may be waiting to see whether you follow through. Those are different situations, and they call for different next steps.

The useful move is to make the next plan small enough that saying yes costs her nothing. A specific time, a specific place, and a reason tied to something you already talked about gives her something concrete to respond to. You are not asking her to design the evening, just to decide whether she wants to be there. Keep the logistics simple and make the choice easy to answer in one word.

If you have been the only one proposing, it can help to hand her a small piece of the planning, such as letting her pick between two places or choose the day. This is a real invitation for her to put something in, and it shows you are not trying to control the whole plan. Be careful not to make this sound like a complaint or a hint. Offering a choice is a different message from asking why she has not suggested anything.

Pay attention to whether the planning shifts at all after you make a proposal she likes. Plenty of people take turns once a plan is in motion, because it becomes obvious that the other person is in. If she is still never initiating after a few successful dates, that is worth a direct but light question in the chat, something like asking what kind of plan she would enjoy, so she has an easy opening to share.

If you are consistently the only one doing the work after several weeks and she is still lukewarm about booking anything, stop leaning in harder. Keep your proposals clear and stop chasing the ones she does not pick up. Her response to a concrete plan is the most reliable signal you will get, and if it stays vague, the answer is usually that she is not invested enough to meet you halfway.