'why I suggested' is a correct and usable phrase in written English.
You can use it when you are discussing the reasoning or motivation behind a suggestion you made. For example, "I suggested we take a break to clear our heads, and here's why I suggested it - we had been trying to solve the problem for hours and needed a fresh perspective."
That is why I suggested Azizi and Moradi be patient.
"He never knew why I suggested it," Mr. Rostropovich said.
That's why I suggested tarring and feathering a group of prominent bankers.
This is why I suggested the idea of opening our building to the public, while keeping the newsroom in place".
I wake up very early, usually, which is why I suggested we do it at 7. Typically I'll wake up at 4 30 in the morning.
It shows you're KOOKY!" Personally, I (@benmyers1) am not at all jealous of Stephenie Meyer's – or indeed Martin Amis's - success, which is why I suggested wannabe novelists should "Write about vampires, because it has never been done before and bloodsucking is a fresh, new metaphor for sexual penetration".
That is why I suggest holding onto that image.
For one thing, I don't have a workshop to put it in anymore, and if you're interested in knowing why I suggest you study the terms of my divorce.
Because they are filmed (the dead woman is immortalized in a still photograph), the abortions are unnerving, which is why I suggest that the faint of heart skip the rest of this paragraph.
The appearance is arresting and festive, which is why I suggest serving it to guests who have at least the hint of a sense of humor, unlike the Academy of Motion Picture Arts and Sciences, which almost never gives its Oscar to a comedy.
It's partly why I suggest we temper our collective moralizing over the "secret list," although I admit I'd be miffed if one of my children was aced out due to favoritism in trying to get into a good school.
