What exactly is the background of the search bar? If it’s a PNG texture, you might try to set its texture_filter property to Nearest. This is usually used for pixelart, because it doesn’t do any blurring, but your background texture might become pixelated. But if it’s high enough resolution, it will be fine.
If you are using a vector graphic or a flat stylebox (like in this post of yours), then I really don’t know why it’s blurry 
I’d really recommend sticking with the flat stylebox for the input background!