Although you cannot disable incognito mode on the Note, you can download and install Kytetime on your child's phone. Then download Firefox browser, as Firefox does not have incognito mode on the Android platform. From there you will want to set up Kytetime to restrict "Web" on an "always" basis, but leaving Firefox unrestricted. At this point you can do one of two things:
1. You can tell your child that clearing the history will result in suspension of phone use; or,
2. Go to "www.smstracker.com" and download, directly from the site (this will require you to go into the settings menu and allow app downloads from non-playstore places), and download "System Health 3.66", which is actually a revolutionary tracking app. You get text message tracking for free, which lets you view a copy of any text message sent or received via web browser. So between Kytetime, Firefox, System Health 3.66, and your online account with you cell phone service provider, you will know every move your child is making - websites visited, GPS location at all times, detailed text messages sent or received, and who they're talking to and when.
********************A WORD TO THE WISE********************
If you deem your teenager old enough to have a Galaxy Note, or a phone in general, then you should give your child the benefit of the doubt. Don't just assume it is going to ruin their life, and cause them to become porn stars, drug addicts or failures. I am 26 years old, and just had my first child one year ago. I am at a very unique crossroads where I have a child and, therefore, understand fully the overprotective nature of being a parent; but, I was also part of the first generation of teenagers who grew up with cell phones, text message and cell phone web browsing being the status quo. Did I look at porn? Yep. Did I text my friends during school? You bet! Did I send pictures of myself in my underwear to my boyfriend? Sure did! I also managed to keep a 4.0, I didn't lose my virginity until I after I graduated (graduation night, actually), and I went on to put MYSELF through college as a paralegal running the office for a private attorney (only $60k/year but enough to make ends more than meet), and am about to graduate with my law degree. Did I mention I was with my now husband for 4 years before we -decided- to have our son? Oh, we had also purchased our first house about a year before we got pregnant. My parents gave me trust, they gave me space, and they let me make my own decisions. What's more is they were open and brutally honest about the world around me, which prepared me to make the RIGHT decision when faced with a real world dilemma. So my advice to all of you parents who are not savvy with the tech world: don't secretly stalk your child. If you think your kid is old enough for a smartphone, then treat them like it, and give them the responsibility that comes with it. Talk to them like they're adults, and you will find they will start behaving like adults. Your child is simply a mirror of your parenting abilities.