How do you handle these situations?
By process I mean you need to define the work at a level of detail sufficient to let you estimate it reasonably accurately. This will also give you a milestone guideline to communicate to the client. And if the client comes back and makes a change, you adjust the date and milestone. I got to the point that unless it was like a simple spelling change, I'd always add 1 day minimum to the schedule.
Lastly you need to add in a factor (multiplier), I used to use 3 different factors complexity, client type and risk. Complexity was project complexity. A marketing website with static content might be a 1. A complex algorithm implemented in a low level language might be a 3 etc. Client Type is all about the client, e.g. small business (new business), medium business, large business, enterprise and then add also in who is the person on their side making final decisions. A super experienced person from a mid-sized company who seemed on the ball might be a 1 (no extra factor), but a new entrepreneur or startup founder might be a 2-3 depending on how I felt they were. Lastly, risk, risk is all about the level of risk you feel this project is to you. A high risk project would get a high multiple, low risk gets a 1.
Using this system I'd literally sometimes have a schedule that the estimate would be 4-6x longer then my gut would've initially said. If things go great then awesome, you deliver early and everyone is happy. But more times then not this saved my butt and we were on-time or slightly early. You handle the objection by telling them you are not the guy that is going to blow smoke up their ass, you are giving them a real schedule that you can keep and you will be aggressively trying to be done early.
But remember to always "estimate" the same way, whatever that way is. Just add an artificial factor that you don't think should be there.
If sometimes you end up really being in line with your (noninflated) estimate, feel free to lower the price for your clients: they will appreciate it and have even more trust in you in the future.
Read and ponder the meaning of the first five panels here represent: https://dilbert.com/strip/1995-01-22