I record screencasts to present deliverables and training materials to clients and colleagues and use iShowU Pro to do this. It's a great app at a fair price, but getting the settings dialed-in so that you can produce good quality screencasts that are reasonable file sizes takes some tinkering. This is how I do it.
As the app will warn you upon initial launch, you'll need to grant it access to control your computer here:
You'll probably want to install Soundflower as well so that you can capture audio from applications:
Then you'll want to get your capture settings applied. I use my Apple-issued headphones connected to my computer and have the 3 tabs configured like this:
Note that the screenshots here show a resolution that is just shy of 1080p - this is because I was off by 2 pixels in either direction while aiming for what is described in the Positioning section below.
I open this 1080p.png file, then position my browser window within the white image area to fit it manually to 1920 x 1080 so that it looks like this:
Note that in the case of Safari, it's best to open a second tab before positioning the window as the browser's height will change when you do so.
With your browser window sized as you'd like it, then you need to tell iShowU Pro to capture only that area - select the Choose button at the top left, hit the space key to change modes to selecting a window and select your browser window.