We study experimentally accessible lower bounds on entanglement measures based on entropic uncertainty relations. Experimentally quantifying entanglement is highly desired for applications of quantum simulation experiments to fundamental questions, e.g. in quantum statistical mechanics and condensed matter physics. At the same time it poses a significant challenge as the evaluation of entanglement measures typically requires the full reconstruction of the quantum state, which is extremely costly in terms of measurement statistics. We derive an improved entanglement bound for bipartite systems, which requires measuring joint probability distributions in only two different measurement settings per subsystem, and demonstrate its power by applying it to currently operational experimental setups for quantum simulation with cold atoms. Examining the tightness of the derived entanglement bound, we find that the set of pure states for which our relation is tight is strongly restricted. We show that for measurements in mutually unbiased bases the only pure states that saturate the bound are maximally entangled states on a subspace of the bipartite Hilbert space (this includes product states). We further show that our relation can also be employed for entanglement detection using generalized measurements, i.e. when not all measurement outcomes can be resolved individually by the detector. In addition, the impact of local conserved quantities on the detectable entanglement is discussed.