My impression as a learner is that 弄 functions as a unspecified verb. Basically, we take a sentence and replace a specific verb (or multiple verbs) by the unspecific 弄.
弄 is typically used within a result complement such as 弄到, 弄坏, 弄丢, 弄清楚, and so on. I'll focus on this common usage. In this way 弄 offers a short-cut to the result without specifying the verb which caused it. (It's also used in 弄得怎么样 and 怎么弄.)
In English, an unspecified verb would normally be phrased "to do", which is sometimes an appropriate translation. We'd use it in English in the phrase "get my hair done" in place of "get my hair cut", "get my hair dyed", etc.
Why do this? This is probably an incomplete list but here's some reasons...
1. Ignorance of which verb is correct
Let's start with this example:
The guest broke [by bashing] the TV
In Chinese, the verb is 打 = "hit" and the result is 坏 = "break".
Now suppose we didn't know how the guest broke the TV, only that they broke the TV. Maybe they spilled a drink on the TV. Maybe they installed some software on it that broke it. Since we don't know what verb to use, we might use the unspecific 弄:
version 1: 客人弄坏了电视机
version 2: 客人把电视机弄坏了[src]
The guest broke [in some unknown way] the TV
Note: It's common to use 把 in conjunction with 弄 as above (see: Using "ba" sentences).
Boy lost [in some unknown way] a public bike; asked for 25,500 yuan in compensation
Kid inattentively blinded [in some unknown way] another person's eyes
2. Multiple ways of doing something
This is similar to how "done" is used in "get your hair done" in English. If you go to a hairdresser, they don't immediately know what verb is appropriate for your hair. Maybe "dye" or "cut" or some other verb applies. Maybe multiple verbs are needed (e.g. the customer wants their hair both cut and dyed). For example:
How to do hair, both good looking and simple: photos
Solved! Why Wei Shihao wants to do this kind of new hairstyle.
Other examples are:
How to be able to make clear [in some way] what you want?
When leaving office, [you] must understand [in some way] these four points, or else it will result in more job-hopping and more bad luck
We "make something clear" or "understand" = "make something understood (to yourself)", but there's not a singly way of achieving this.
3. To encompass all possibly ways of doing something
Let's start here:
Don't damage your clothes by washing them.
Don't damage your clothes by wearing them out.
Don't damage your clothes by cutting them.
We can say all of these (and more) at once using:
Don't damage [in any way] your clothes.
In this stadium, I played my last match, don't make me cry [in any way], because I am now happy
4. To withhold this information
Sha Yi made [in some unspecified way] Guo Jingmeng cry
You need to read the article to understand how Sha Yi made Guo Jingmeng cry.
You've already messed up [in some unspecified way] my heart, when will you mess up [in some unspecified way] my bed?
So let's try it on the given examples:
Don't make the child cry.
I found this used here.
Loosen the belt.
The closest I found online is:
Newly-bought T-shirt sleeves are a little tight, how to loosen them?
There is one more problem that has yet to be clarified.
I found this used here.