What you are explaining is the actual Screen Reader behavior. When an image has an alt value, the screen reader will read it. If you want the image to not be accesible by the screen reader here are your options:

1. Remove the value in the alt text.

<img src="..." alt="">

2. Add the property role with the value presentation.

<img src="..." role="presentation">

Notes to consider:

• if you require an alt value, go with the second option.
• use the second option for decorative images. Images without meaning to the user.
• if your image has a meaning or relation to the content around it, you require the alt value and by default the Screen Reader will access it.