This is not possible OOTB. You need to create a custom new form.
Usually new form URL for specific content type in is in below format:
0x01001F4E17A590CC8E4998BE646AEA065BFE is Content type ID and
RootFolder is URL of your list.
Follow below steps:
- Create a drop-down in your custom form to select content type which will have display text as a content type name and value as a content type ID.
- On selection/change of Content type drop-down build the new form URL for that particular content type in above format.
- Redirect your Page to the newly build URL.
- Also, read content type from URL. If Content type Id is available then set your Content type drop-down value with content type from URL.
Additionally, OOTB you can always select the Content Type while creating new list item like below: