Необходимо после 6 элемента массива добавить row и изменить колонки на div class="col-md-6 col-xl-3 institute-preview-wrap"
<div class="row">
<?foreach($arResult["ITEMS"] as $arItem):?>
<?$this->AddEditAction($arItem['ID'], $arItem['EDIT_LINK'], CIBlock::GetArrayByID($arItem["IBLOCK_ID"], "ELEMENT_EDIT"));
$this->AddDeleteAction($arItem['ID'], $arItem['DELETE_LINK'], CIBlock::GetArrayByID($arItem["IBLOCK_ID"], "ELEMENT_DELETE"), array("CONFIRM" => GetMessage('CT_BNL_ELEMENT_DELETE_CONFIRM')));?>
<div class="col-md-6 col-xl-4 institute-preview-wrap">
<a href="<?echo $arItem["DETAIL_PAGE_URL"]?>" class="institute-preview">
<h3 class="institute-preview__title"><?echo $arItem["NAME"]?></h3>
<div class="institute-preview__description"><? echo $arItem["DISPLAY_PROPERTIES"]["ABBREVIATION"]["DISPLAY_VALUE"];?></div>
<div class="institute-preview__contact-wrap">
<i class="fas fa-map-marker"></i>
<span class="institute-preview__address"><? echo $arItem["DISPLAY_PROPERTIES"]["ADDRESS"]["DISPLAY_VALUE"];?></span>
</div>
<?if ($arItem["DISPLAY_PROPERTIES"]["PHONE"]["DISPLAY_VALUE"]):?>
<div class="institute-preview__contact-wrap">
<i class="fas fa-phone"></i>
<span class="institute-preview__phone1"><? echo $arItem["DISPLAY_PROPERTIES"]["PHONE"]["DISPLAY_VALUE"];?></span>
</div>
<?endif;?>
<?if ($arItem["DISPLAY_PROPERTIES"]["EMAIL"]["DISPLAY_VALUE"]):?>
<div class="institute-preview__contact-wrap">
<i class="far fa-envelope"></i>
<span class="institute-preview__email"><? echo $arItem["DISPLAY_PROPERTIES"]["EMAIL"]["DISPLAY_VALUE"];?></span>
</div>
<?endif;?>
<div class="institute-preview__established">Основан в <? echo $arItem["DISPLAY_PROPERTIES"]["YEAR_FOUNDATION"]["DISPLAY_VALUE"];?> году</div>
<div class="institute-preview__img-wrap">
<?$imgIconInst = CFile::GetPath($arItem["PROPERTIES"]["ICON"]["VALUE"]);?>
<img src="<?echo $imgIconInst;?>" alt="" class="institute-preview__img">
</div>
</a>
</div>
<?endforeach;?>
</div>