I have a lot of sympathy. My DD1 (also 4) sounds much like this. We used to joke that she had started the terrible twos early, but we never expected the tantrums still to be going when she was rising 5. It's like she is two people. Most of the time she is funny, kind, considerate, clever etc, but then she flips and kicks and hits and screams, bangs doors and throws things. She is BIG now and it is hard for me to physically restrain her from doing damage (to me / the house). Somtimes I am literally at my wits end. Undoubtably the best thing to do is to leave her alone to scream it out as any intervention (reasoning / hugs / shouting) only seem to add fuel to the fire. The problem is once something has set her off it can take well over an hour before the heat of the tantrum and screaming abates and she is left worn out and ready for cuddles (except by that stage you can be certain that the last thing I feel like doing is cuddling her).
I'm not sure what advice to give you as if I knew I would be doing it myself. She is definitely worse when she is tired (but she's always tired - has just started reception and finds it exhausting). I really do try to avoid known flash points and recently have been keeping a note book to record when and why it all kicks off to see if I can identify a pattern. I try and try not to let it get to me and certainly if I can remain calm everything is a million times better. But I can't always do that. I get tired and grumpy too and sometimes I just lose it and find myself shouting and screaming back at her which is a disaster.
I do think that if I can see the situation approaching (i.e. tired, fractious etc) and can get her on my knee for a cuddle and give her a reasonably sugary snack then she might make it through. I know this goes against the grain and certainly like many children she goes crazy after cake / chocolate etc, but there is definitely a point in the afternoon when she is back from school and blood sugar levels are low and she is liable to FLIP, but a well timed bit of fruit and a biscuit can often save the day.
My bath time was pretty much like yours on the screaming front this evening to be honest. Although she was very happy to have her bath and afterwards was on my knee cuddling as I dried her she then wanted me to carry her to her bedroom wrapped in her towel (as I had just done with her 2 yr old sister and baby brother). The thing is I can't carry her any more, she's too heavy, so she went absolutely mad and screamed in the bathroom for half an hour. It's just so silly. OK she was jealous - but we'd been having such a nice time, so happy, so relaxed, cuddling etc and then just like that she went mad. Unfortunately I reacted badly and got pretty cross - I wish I hadn't, it's such a waste of energy. I know she was jealous, but I can't predict where her jealousy might spring from and don't think to watch my every action.
She is such hard work and it is very tough. I long for it to improve and must have read a million books on how to deal with this and although most of what they say makes sense and I try to put it into practice still it goes on and on..... It's just so difficult.
SO what should you have done.... i don't know, maybe the threat should be more immediate so it can be carried out there and then and cause and effect is obvious (e.g. if no bath then no bedtime story). Probably it's best to try and hold off threats altogether to be honest - by the time my DD is in that state she can't really recognise or care about what is being threatened. What always works best for us is if I can let her scream it out and vent the fury, then give her a hug without any particular comment on the tantrum and continue with whatever I wanted her to do in the first place. It's so time consuming though and I don't always have the patience. You say you tried to ignore her - but you didn't ignore her. The trick if you can manage it is really and truly to ignore her, it definitely is the best thing.
My mantra (which I try and remind myself of when I'm getting more and more angry and frustrated) is 'she needs your love most when she deserves it least'