The getType method you are calling directly returns a DisplayType value rather than a String so change your code to:

DisplayType displayType = record.getDescribe().fields
    .getMap()
    .get(fieldName)
    .getDescribe()
    .getType();

See the DescribeFieldResult documentation for more detail.

The getType method you are calling directly returns a DisplayType value rather than a String so change your code to:

DisplayType displayType = record.getDescribe().fields
    .getMap()
    .get(fieldName)
    .getDescribe()
    .getType();

See the DescribeFieldResult documentation for more detail.

PS

In response to the first comment, in your approach, you are running Type.forName on one of the enum values e.g. DisplayType.Picklist not the enum type DisplayType. So it's not going to work even if Apex treated enum as a special kind of class.

Also note that trying to cast a field value to a DisplayType won't work. For types other than strings you will likely need to write your own type conversion code based on the DisplayType or likely better based on the SOAPType that is also available from a describe call.

Given this complexity do consider if your overall approach here is appropriate.