It sounds like you’re doing something like this currently:
<script type='text/x-handlebars'>
{{#if twitchLink}}
[...]
{{/if}}
</script>
<script type='text/x-handlebars'>
{{#if steamLink}}
[...]
{{/if}}
</script>
If instead you do this:
<script type='text/x-handlebars'>
{{#if twitchLink}}
[...]
{{/if}}
{{#if steamLink}}
[...]
{{/if}}
</script>
You can then simply set the style attribute on the outermost div or h3 to display: inline-block;, so, something like this:
{{#if twitchLink}}
<div class="public-user-fields" style="display: inline-block;">
[...]
(Or just use elements which are naturally inline).
If you’re looking for more inspiration, this gist should do what you want:
Change the object on line 4 of head.html to change the row of icons, and on line 50 to change the icon & text combo which appears next to the location/site.