Quantum measurements are not discontinuous, instantaneous processes but rather determined by the continous evolution of the joint Hamiltonian comprising the system being measured and the measurement system. You can perform partial measurements of a quantum state and even exert feedback on the quantum system to keep it at a given state (so-called quantum feedback). The outcome of a given measurement is only random because we are part of the measurement system and become entangled with it, which in combination with decoherence - due to the near-infinite number of degrees of freedom in the measurement system - leads to our experience of "quantum jumps". All of this can be explained with the Schrödinger equation alone, without the need to invoke the principle of wave-function collapse.

I don't like theories that require such a wave-function collapse mechanism in order to get rid off all the extra "worlds" because they would require a physically plausible mechanism that governs this collapse and the breaking of time-reversal symmetry that it creates. It is also hard to think about a scaling mechanism that would govern the wave-function collapse: If we imagine that we will one day be able to build large quantum computers with many millions or even billions of quantum bits we will be able to couple e.g. a single qubit to the computer and perform a pseudo-random (but entirely deterministic) range of operations on the other qubits. Should we assume that at some point the wavefunction of the single qubit that is coupled to the many other qubits will collapse? In that case we should not be able to reverse the qubit to its initial state by reversing all operations performed on the other qubits. At which scale should this happen then? If we could eventually scale up the quantum computer to comprise a near infinite amount of qubits in such a way that it's able to run a simulation of a toy universe with a conscious observer in it, would that be enough to induce wave-function collapse (even if we can still reverse the deterministic gate sequence at any time)? In such a case, for each qubit state there would also be a version of the observer, each seeing a different "collapsed" wave-function of the qubit. However, we could still reverse the gate sequence of the computer at some point and bring it back to its original state, uncollapsing the wavefunction of the qubit in the process. Now, we would of course also reverse the state of the observer(s) to their/its initial one. The question is, can we restore the state of the single qubit to an uncollapsed quantum state while keeping the state of the observers untouched? And if we could do it, what would it mean for the observer(s), will we collapse their wave function in the process? Given infinitely precise control over our quantum computer we should actually be able to do it since the evolution of the quantum system is still deterministic. The more interesting question is if this reversal of the single qubit state will automatically lead to a collapse of the two observer(s) back to a single one as well (here I'm not sure but this should be calculable).

In any case the conscious observer in our quantum computer would have no way (as far as I can think of) to determine whether it lives in a many-worlds quantum universe or one where wave functions collapse. We, on the other hand, would know that its a many-worlds quantum universe because we just execute a reversible, deterministic gate-sequence on the computer.

Sorry for rambling, my point is that it's a very metaphysical and (mostly) irrelevant question for most physicists, and it's unlikely that we will really come up with a way of deciding this question as we are part of the system that we are trying to investigate. Answering the questions for a quantum world that we simulate in a powerful computer might be possible, but it won't tell us much about our own universe. I personally just prefer the many-worlds hypothesis since it seems more elegant and simple and does not require coming up with a new mechanism for destroying time-reversal symmetric in quantum mechanics.