How to mask a URL in PowerBI

There is actually a way to do this.

In a table (data source), you must have these two fields: the url link and the name (aka alias) of what you want it to be displayed as.

In the table (visual), add the alias field as you want it to be displayed. Go to "Format Visual" under the Visualizations tab and scroll down to "Specific Column". Choose the alias column, making sure "Apply to values" is set to "ON". 

Scroll down a bit further to "Cell elements" and select the same column again. Go to "Web URL" and set that to "ON". Then, click on the "fx" button.

A new window pops up. Under "Format style" select "Field value" from the drop down and apply to either "Values only" or whatever applies to your case.

Under "What field should we base this on?", select the url field you had in your data source table. Set "Summarization" to "First", then click "OK".

Note: This can work with static/string fields as well as conditional formatted ones. For example, if you want the name displayed to change according to what the url says, or any other criteria, you can create the DAX function in the "Data" view, then apply the steps above to have url aliases that change dynamically.