Finding dense subgraphs is a core problem with numerous graph mining applications such as community detection in social networks and anomaly detection. However, in many real-world networks connections are not equal. One way to label edges as either strong or weak is to use strong triadic closure~(STC). Here, if one node connects strongly with two other nodes, then those two nodes should be connected at least with a weak edge. STC-labelings are not unique and finding the maximum number of strong edges is NP-hard. In this paper, we apply STC to dense subgraph discovery. More formally, our score for a given subgraph is the ratio between the sum of the number of strong edges and weak edges, weighted by a user parameter $λ$, and the number of nodes of the subgraph. Our goal is to find a subgraph and an STC-labeling maximizing the score. We show that for $λ= 1$, our problem is equivalent to finding the densest subgraph, while for $λ= 0$, our problem is equivalent to finding the largest clique, making our problem NP-hard. We propose an exact algorithm based on integer linear programming and four practical polynomial-time heuristics. We present an extensive experimental study that shows that our algorithms can find the ground truth in synthetic datasets and run efficiently in real-world datasets.