If you are familiar with PyQGIS you could do something like this (QGIS 3.4.1):
# 'lines' is a polyline vector layer with one empty attribute field.
# It could have more fields but remember to adjust the field index
# in the attributes dictionary - {feature_number:{field_index:attribute_value}}
l = QgsProject.instance().mapLayersByName('lines')[0]
feats = l.getFeatures()
count = []
for f in feats:
l.selectByIds([f.id()])
j = processing.run('native:saveselectedfeatures',{'INPUT':l,'OUTPUT':'memory:target_line'})
processing.run('native:selectbylocation', {'INPUT':l,'PREDICATE':7,'INTERSECT':j['OUTPUT'],'METHOD':0})
s = processing.run('native:saveselectedfeatures',{'INPUT':l,'OUTPUT':'memory:selection'})
t = len(list(s['OUTPUT'].getFeatures()))
r = {f.id():{0:t}}
count.append(r)
l_prov = l.dataProvider()
for i in count:
l_prov.changeAttributeValues(i)

Bear in mind that it counts intersecting features, not number of intersections.