The phrase "suggest you stop" is correct and it can be used in written English.
Example: I suggest you stop and take a break to clear your mind.
Once you've decided, may we suggest you stop by our Rereading Stephen King series?
Finally he may suggest you stop fretting: new technologies will soon fill the glass up anyway.
Now comes the deflating part, and I suggest you stop reading if you're easily scarred by disappointment.
If you start acting a bit like Nicholson's character does in the above clip, we suggest you stop and take a break.
However, from your longer letter I can see that it bothers you, so can I suggest you stop the machine and scrape the sides of the bowl down constantly with a rubber spatula during the creaming of the butter and sugar and get it really light and fluffy before continuing.
I highly suggest you stop by Jo's for lunch.
I'm not suggesting you stop responding to emails.
Use it, and some lawyers from The Coca-Cola Company will send you a letter politely suggesting you stop using their trademarked brand name.
I'm not suggesting you stop getting things done or moving projects forward.
"What if we lose this customer post purchase of the company?" To be clear, I'm not suggesting you stop selling to this customer, of course not.
If an agency asks for an up-front fee to help you with this process or suggests you stop paying your loans and pay them instead to handle the consolidation, this should raise a red flag for you.
