Since you are using a custom template, the quicker way is just to use the replace() filter.
{{ content.field_testing|replace({'<br>': '', '<br />': ''}) }}
The field formatter used in the case of a plain text field doesn't have settings, so you cannot decide if the formatter should output the <br /> tags you want to remove. It would probably make sense in few cases, especially if you need to replace those <br /> with other text, but somebody could argue that if you don't need them, you could just avoid entering new lines in the field.
The alternative would be implementing a different field formatter, but this is not as quicker as using a filter. If you just need this for this exact purpose, it sounds a bit excessive.