A logo can look fine on a white page and grow a box around it somewhere else. Usually the empty area was filled during export, or the destination is showing it over a different background. Before changing the artwork, check whether the saved file still has transparency.
Transparency is stored with the image. It isn't the same as white pixels around a shape.
A small example you can check
The sample PNG is 320 by 180 pixels. It has a solid blue shape, a red circle with partial transparency, and a fully transparent corner. There's also an SVG version for trying the vector export controls. Both are generated, so you don't need one of your own photos.
Open the PNG in Image Converter. Leave the output as PNG, then save the converted file. Change Convert to to JPG and save that copy too. Compare the files on a light background and a dark one. The blue shape is opaque in both. The difference shows up in the empty area and the translucent circle.
In Chromium 146 on Windows, the converted PNG kept its transparent corner and the JPG corner turned opaque black. The converter draws onto a clear canvas and doesn't paint a background before writing JPG. That's how this tool behaves. Other converters don't necessarily pick black.
The two outputs also show why file size shouldn't make this decision for you. For this sample, the PNG export was 2,517 bytes and the JPG at the default 92% quality was 3,936 bytes. A JPG extension doesn't guarantee a smaller file, especially for a simple graphic. Other browsers and pictures can produce different sizes.
Give JPG a deliberate background
JPG can't store transparency. If you need a specific background for an SVG, open the sample in SVG To Raster, select JPG, and choose Background color. It starts at white. The exporter fills the canvas before drawing the artwork, so you're choosing the surface the translucent pixels blend into.
For PNG or WebP, Transparent background controls whether that fill is used. Keep it on when the destination needs a clear background. Turn it off when you want the export to have its own background color. That color then becomes part of the picture instead of something the receiving page supplies.
Image Converter doesn't have a background color option for existing bitmaps. Use PNG or WebP to keep the alpha, or add the background in an image editor before converting to JPG. Renaming the file won't add transparency or pick a fill.
Partial transparency matters at the edges
A pixel can be partly transparent. That lets a curved edge blend into the surface behind it instead of looking like a staircase. The red circle in the sample shows this. It looks different over white, black, and blue even though the file hasn't changed.
A checkerboard is only a preview background. It helps you tell empty space from white pixels, but it shouldn't end up baked into a transparent export. A logo that looks right on white might also show a pale fringe on a dark page. Check it on the destination background at the size it'll be shown.
Save a copy that fits the destination
PNG is a good starting point for this kind of graphic. WebP can keep transparency too, if the app you're sending it to accepts WebP. The image format guide covers that choice in more detail.
Use Image Compare for the visible differences, then open the downloaded file in its destination. Keep the transparent source. Flattening onto white and saving the result as PNG later won't bring the empty area back. By then the white is part of the image.