"My nose is running a lot."
"はな水" (鼻水) is a word meaning "nasal mucus". This sentence literally says "A lot of nasal mucus is flowing." (出る can be used to talk about flow of fluid from a body). Why it's in the plain form, I'm unsure, but my best guess it that it's because it's the speaker's body, as opposed to someone else's, performing the action. Perhaps one would use the polite conjugation if one was talking about someone else's nasal mucus flowing.
That said, はな水 is a distinct word, and should be treated as such in the program if we are to be expected to use it. One could say that its meaning is easy to discern, but the same could be said about quite a few words in the course. As things stand, it's not listed in the lesson's words and so I doubt it will be something that's kept track of in our learned words.
From other materials I have studied, I get the impression たくさん and other similar adverbs of degree are often used straight before the verb and Japanese sentences have somewhat fluid structure as long as the adjective or verb comes at the end of the clause or sentence. However, this duolingo course tends to be very rigid and allow only one way of saying things and other, oftem correct ways, of saying them are deemed automatically incorrect. Hopefully this is something duolingo will correct over time as it has to some degree with its courses for some other languages. That said, I feel that this sentence is not very correct and reads very strange to me. If a native speaker reads this, could you tell us if my observation is correct or not, and if this sentence is how a native would say it?
A native speaker here. Your observation is correct. Grammatically the adverbs come straight before the verbs but 倒置法inversion can occur when placing stress on certain parts of a sentence. 「たくさんはなみずがでます。」
Some notes below:
Speech: Suppose if you are at a clinic and doctor is asking your symptom, you would be cordial & talk politely, and use です、ます。
Word order: symptom is runny nose. Bring たくさん to the beginning of the sentence accentuates the severity. The opposite is すこし鼻水がでます。
鼻水：it is true that you can also sayはなが出る、in this case the correct Kanji is 洟（はな）
Hope the above helps.
The order of this sentence is so weird compared to everything else. I would've expected something along the lines of はながたくさん水を出る
Thinking of it as "lots of nose water is coming out" makes it a little bit easier to remember the order even though it's a bit of a weird phrase, if anyone is having similar troubles to me
IMO it makes sense if you think of たくさん as referring less to the actual volume of snottage than to the overall situation—e.g., maybe your nose isn't producing all that much but has been running frequently.
"Man, my nose has been running a ton!"
"Looks like a trickle to me."
"But for DAYS!"
So, the situation is that 鼻水が出る, and it's true a lot.
Similarly, maybe, in とても頭がいい, とても intensifies the 頭がいい situation, rather than the head. Maybe not the best example (or even correct), but see what I mean?