Apple:

Apple is expanding its Independent Repair Provider Program with additional options for customers to access repair services. The industry-leading program enables businesses of all sizes to offer repairs on iPhone using genuine Apple parts, which ensures safety and quality. Following the launch of the program in the US last fall, over 140 independent repair companies have joined with over 700 new US locations now available to customers, and businesses in Europe and Canada can now sign up.

And:

Since the launch of the Independent Repair Provider Program last fall, there are now over 700 Independent Repair Provider locations across the US providing out-of-warranty service for iPhone.

Genuine Apple parts is key here, especially where screen replacement is concerned.

You can verify that your local shop has access to genuine parts and repair resources on this official Apple page.