You can definitely use HDMI to USB-C. This is the one I purchased, works flawlessly. You have two USB3.0 on the left side for Mouse/KB. However, I use a USB Hub that I position a lot closer to my wireless mouse to ensure good response rates for gaming.

https://www.amazon.com/Belkin-F2CU038btBLK-USB-C-Type-Adapter/dp/B01I2TTCTK

Please provide the C-to-HDMI adapter you are using here. It could be that that piece is buggy.

Also, please provide the make and model of your GPU.

But still very strange that only certain websites give me issues. Someone below mentioned HDCP

It's actually not strange, and HDCP perfectly explains your symptoms, if the site you are seeing this on most often is Amazon Prime Video streaming.

HDCP is a digital rights management (DRM) technology that encrypts video data so that it cannot be easily copied by an in-between device sitting between your computer and the monitor.

It's enabled only on a case-by-case basis. Basically, normal web sites without paid-video sort of content has no reason to turn it on, but the actual video pages for Amazon Prime Video streaming, Hulu, and Netflix definitely do.

What's probably happening here is either your GPU, your C-to-HDMI adapter, or your monitor is screwing up HDCP somehow in this setup.

You can try to find another C-to-HDMI adapter that says in its spec sheet "HDCP" compliance. As an example, here's Belkin's: https://www.amazon.com/Belkin-F2CU038btBLK-USB-C-Type-Adapter/dp/B01I2TTCTK

You can see in its product description it supports HDCP 2.2.

Play protected content from iTunes/Netflix with support for HDCP 2.2