Looking on the internet deeply has found these results:

suggest you do is the most popular phrase on the web.

suggest you to do 125,100 results on the web	suggest you do 146,800,000 results on the web More popular!
Some examples and use cases from the internet:	Some examples and use cases from the internet:
Whatever you planning to do, Centauri I suggest you to do it quickly. I don't think he suggests you had anything to do with this. Well, may I suggest to you that you do not underestimate your present company. I suggest you do something to discourage her. Now I keep investigating, and since I know it's futile to suggest you don't do the same, do me a favour. Now another exercise that I suggest you to do once a day. Because this whole thing of doing what you want me to do and what you suggest me to do and you trying to fix my problems, it's not working. So I suggest you try to do the same.	Here's what I suggest you do. I'm following orders. I suggest you do the same. No, you made a deal to put away the person responsible for ordering these killings, and I suggest you do that while you still can. I suggest you do something to discourage her. I suggest you do something, that is very interesting... I suggest you do the same. And I suggest you do the same... I suggest you do as he says.

suggest you to do

125,100 results on the web

suggest you do

146,800,000 results on the web

More popular!

Whatever you planning to do, Centauri I suggest you to do it quickly.
I don't think he suggests you had anything to do with this.
Well, may I suggest to you that you do not underestimate your present company.
I suggest you do something to discourage her.
Now I keep investigating, and since I know it's futile to suggest you don't do the same, do me a favour.
Now another exercise that I suggest you to do once a day.
Because this whole thing of doing what you want me to do and what you suggest me to do and you trying to fix my problems, it's not working.
So I suggest you try to do the same.

Here's what I suggest you do.
I'm following orders. I suggest you do the same.
No, you made a deal to put away the person responsible for ordering these killings, and I suggest you do that while you still can.
I suggest you do something to discourage her.
I suggest you do something, that is very interesting...
I suggest you do the same.
And I suggest you do the same...
I suggest you do as he says.