The "little sponge" crap that we are constantly told about multilingualism drives me mad. (I have a fully bilingual child, by the way.)
Children's ability to learn languages to fluency (more or less) is remarkable BUT:
a) They need a lot of exposure to get to full native level with any language. The ballpark figure one generally hears is that to become a bona fide native speaker in any language, you need to be exposed to it for at least 30% of your waking hours.
b) Even within the "native speaker" scope, individuals' ability will vary quite a lot when it comes to the "higher-level" stuff--such as second tier vocabulary (the more specialized and literary vocab, basically), written language skills, and ability to understand complicated written text.
c) Children may pick up playground level fluency in less than a year but it takes years to get fully to grips with a language, esp if we are talking about mastering it to the level of school academics. Again, ballpark figures: about 2-3 years to fully master a spoken language, and another two years on top of that to fully catch up in terms of vocab and written language. (For older children, they may never catch up while they are at school. At four, fortunately, you don't need to worry much about this; she has plenty of time to catch up and there will be other English language learners at her school.)
Four languages sounds like a hell of a lot and I would be concerned that this is going to drive you nuts and also reduce your children's chances of achieving really strong written language and vocab in any one language. I personally recommend three as a max for parents---and even that requires you to be fairly organized about how you balance things. Can you review and consider dropping one language for now?
Your daughter's everyday spoken fluency in English will improve rapidly without tutoring. However, it is quite likely that her vocabulary is more limited than that of other children her own age. To work on vocab with her, tutoring is not really what you need. What you need to be doing is lots of book-sharing.
Make a distinction between learning to read and learning vocab: graded readers like Biff and Chip are good for mastering phonics and decoding, but to improve her vocabulary, you want to be reading to her lots of books that are challenging and have plenty of rich vocabulary. Stuff like Beatrix Potter is good for this age group and is full of wonderful vocab and complex sentence structures. Nonfiction is also very helpful for sharing with children (stuff like Usborne books about science or pyramids or whatever).
Try googling "dialogic reading" and watching a couple of videos about this. This term describes a process by which the parent does not just read, read, read, but slows down and reads at a slower place, stopping to explain new/tricky words and break down long/complex sentences, discuss the story with the child, check that they understand what is going on, encourage them to engage with the book and so on. It massively speeds up vocabulary acquisition in children. I've done this with my daughter since she was young, and she has a really good vocab in English for her age even though her schooling is in another language. You can do this kind of thing with French or her other minority languages too.
If you make a little time to do this each day, her vocab will improve in leaps and bounds. No tutoring required. I have no issue with parents hiring tutors at any age, but they are best used for helping with specific academics skills like phonics or maths.