你没注意到前面的动词是 suggest 吗？

suggest 后面是不能接不定式的复合结构的，尽管汉语可以说“建议某人做某事”，但英语习惯上不能说suggest sb to do sth。如：

他建议我们买台辆新车。

正：He suggested our buying a new car.

正：He suggested that we (should) buy a new car.

误：He suggested us to buy a new car.

不过若换成动词 advise，则完全可说 advise sb to do sth。如:

He advised us to sell the house. 他建议我们把房子卖了。

The doctor advised me to stop smoking. 医生建议我戒烟。

网友的句子：

Where does the woman suggest David go for his next holiday?

Suggest 后面接的是宾语从句，go 前面省略了 should：

Where does the woman suggest David go for his next holiday?

=Where does the woman suggest David should go for his next holiday?