Browsers by default are read. If cookies are enabled and used by the site, the cookies are created on your computer and contain mostly your info, like when you put something in your shopping cart, on the CW online store. It is stored on your computer. Sometimes accessing a browser may put an ad tracker, which tracks your movements to provide demographics to cookie writer. Well I am rambling. Websites do place things on your computer, but only when you access the site or an item on it. Many sites don't know your there until you request something, by clicking on an item in the browser. After it sends you something it ignores you until you come back. Like Verizon there is also a prompter that may pop up after 0-120 seconds of you being idle.
Web browsers do not maintain connection for long. Most forums consider you logged off after 15 minutes of idle and banks or secure sites even sooner.
There is no danger in not logging/closing off of non secure browsers. Always log off of banking type sites.
There are all kinds of nasty stuff, more than pics, on xxx, warez, torent sites.
As someone said above, there is little problem if you have adequate AV, malware software.