Quality loss when exporting motion graphics videos in Premiere Pro
There are 2 things causing this issue:


Chroma sub-sampling. This is explained pretty well in this answer.
Compression artifacts - For example, the mark above the "1"on the email notification is due to the way H.264 compresses data. This is known as "ringing" and is quite common with compression.


Some options for dealing with it include:


Avoid using saturated reds and blues especially in areas of high color contrast
Use a codec or codec settings that don't exhibit ringing (easier said than done when you're uploading to a video service that may recompress it)
Use settings to reduce how noticeable it is, such as turning on motion blur when exporting, so when things move, they exhibit softer edges


Unfortunately, the reality is that TV is terrible for pure motion graphics. The subsampling that occurs makes it impossible to have nice crisp lines in saturated colors. And unfortunately, most of the codecs that work on all platforms are made to deliver TV and movies to users.