'become thinner' is correct and usable in written English.
You could use it to describe a person's physical state, for example: "He had been exercising regularly, and was beginning to become thinner."
The honey should become thinner in consistency without becoming hot.
A similar process also causes women's hair to become thinner.
Others ask her what they can do to become thinner.
As solar panels become thinner and more efficient, less silicon is needed to make each one.
The ice has also become thinner due to the warming oceans.
As we age, she said, "bones become thinner, which leaves them more vulnerable to being fractured".
These deposits become thinner to the east, showing the influence of prevailing westerly winds.
And as computers become thinner, faster and more energy efficient, mobile processors will eventually become crucial for PC makers, too.
"But that system has become thinner and thinner and a lot of students give up before even starting.
I was struck, though, by the advice she dispenses regarding how to become thinner: run more and eat less.
The young migrants who have made the labour force so flexible will become thinner on the ground, and more expensive.
